Well, it's more by a process of elimination that DR remains.  So many of the other English translations have errors in them (many of which also imply theological errors).

I like parts of the Knox Bible.

My big issue with Douay Rheims is the archaic language and complex sentence structure ... by today's standards anyway.  That's just because it's so old.  This makes it difficult for younger children to read and understand.  If you look at the Latin (and original Greek) of the New Testament, it's very simple Greek, with simple syntax and vocabulary.  St. Jerome maintained this same spirit with the Vulgate rendering.  Our Lord taught in a very simple manner.  He never taught like St. Paul tried once at the Athenian agora to produce a great work of oratory (like a Cicero or Demosthenes).  Of course, Our Lord COULD have made these great orators look like idiots if He had chosen to.  While other priests in his day liked to spin eloquent oratory, St. John Vianney spoke very simply and would bring the entire congregation to tears by the end of each sermon.

I had at one point begun my own English translation that would be accessible to children, but I did not make much progress before I realized that I would not have the time to complete it before my own children were fully grown  .

I do believe that the Douay-Rheims could use a little bit of updating in general, even for adults ... but it has to be done by someone with the Catholic faith.

What languages are you fluent or proficient in again (written, not spoken)?

Ancient Greek (written only), Latin (mostly written, but some spoken ... though it's been a while), Hungarian, and of course English.

I used to be able to read German, French, Spanish, and Italian just at a grade-school level (and cobble together a few spoken sentences) ... but it's been over 20 years since I last used those, so I'd be just hacking through it at this point.  Latin actually made Italian, Spanish, and (to a lesser extent) French pretty easy to pick up.  German was its own animal.  I also took a couple semesters of Biblical Hebrew (with Father Mitch Pacwa at Loyola U. Chicago) and could read the Hebrew Old Testament armed with a dictionary, but that's extremely rusty now ... and I'd have to re-learn this.

Very well said Dr. Matto, that is why I have a Haydock DR bible, it is 50% scripture and 50% explanations by the Church Fathers, Doctors, Saints, good priests. I do not think I have ever read much more that a 1/2 page of scripture from a regular bible, it just goes in one ear and out the other. Why ponder about scripture when I have Church Fathers, Doctors, Saints, good priests to explain it to me in the Haydock DR bible?

Yes, for me, the Haydock Bible is the best of them all, and having read it cover to cover only one time a long time ago, I disagree with Matto and recommend undertaking a reading of it in it's entirety, at least one time. 

Fr. Wathen gave an excellent sermon on this subject; The Bible and the Missal, which I found enlightening.   

I have read many of the Bibles to me the Douay Rheims is the most accurate with an explanation the bottom.  Mother of Our Savior also sells Douay Rheims Bibles. A great read is Pope Leo XIII Encyclical Providentissimus Deus On Sacred Scripture. If you want a printed copy go to the Catholic Research Institute. Many of the Encyclicals have been changed by the Vatican to Modern versions extremely distorted.  

If I could only have one Bible, I would choose the D-R (Knox would be second, as it also follows the Vulgate, in most cases). But I would prefer to use the D-R as my principal Bible and reference other translations to shed additional light on certain verses. One must understand that ANY translation reflects, to some degree, the mind and choices of the translator. And with translations from ancient languages into modern languages, that problem is magnified significantly.

Ideally, one would learn Ancient Hebrew, Koine Greek and Ecclesiastical Latin to some degree. Even a rudimentary introduction to those languages would serve to humble you and help you to understand that the meanings of many Biblical words and phrases are intentionally manifold, metaphorical, analogical, allegorical, and not simply literal. I recommend a website like Biblehub.com to seek additional information on the Hebrew and Greek words used in some passages, to understand the range of meanings that they contain.

For me the D-R is the best Bible not necessarily because of the "translation" but rather because the Vulgate is a more reliable ancient source for the base texts. I will give one example from the Book of Tobias, Chapter 6. 

Here is the D-R version (which uses the Vulgate as its source):

14 Then Tobias answered, and said: I hear that she hath been given to seven husbands, and they all died: moreover I have heard, that a devil killed them. 15 Now I am afraid, lest the same thing should happen to me also: and whereas I am the only child of my parents, I should bring down their old age with sorrow to hell. 16 Then the angel Raphael said to him: Hear me, and I will shew thee who they are, over whom the devil can prevail. 17 For they who in such manner receive matrimony, as to shut out God from themselves, and from their mind, and to give themselves to their lust, as the horse and mule, which have not understanding, over them the devil hath power. 18 But thou when thou shalt take her, go into the chamber, and for three days keep thyself continent from her, and give thyself to nothing else but to prayers with her. 19 And on that night lay the liver of the fish on the fire, and the devil shall be driven away. 20 But the second night thou shalt be admitted into the society of the holy Patriarchs. 21 And the third night thou shalt obtain a blessing that sound children may be born of you. 22 And when the third night is past, thou shalt take the virgin with the fear of the Lord, moved rather for love of children than for lust, that in the seed of Abraham thou mayst obtain a blessing in children.

Here is "the same" passage from the "Catholic" New American Bible (which uses questionable sources):

14 But Tobiah said to Raphael in reply, “Brother Azariah, I have heard that she has already been given in marriage to seven husbands, and that they have died in the bridal chamber. On the very night they approached her, they would die. I have also heard it said that it was a demon that killed them. 15 So now I too am afraid of this demon, because it is in love with her and does not harm her; but it kills any man who wishes to come close to her. I am my father’s only child. If I should die, I would bring the life of my father and mother down to their grave in sorrow over me; they have no other son to bury them!” 16 Raphael said to him: “Do you not remember your father’s commands? He ordered you to marry a woman from your own ancestral family. Now listen to me, brother; do not worry about that demon. Take Sarah. I know that tonight she will be given to you as your wife! 17 When you go into the bridal chamber, take some of the fish’s liver and the heart, and place them on the embers intended for incense, and an odor will be given off. 18 As soon as the demon smells the odor, it will flee and never again show itself near her. Then when you are about to have intercourse with her, both of you must first get up to pray. Beg the Lord of heaven that mercy and protection be granted you. Do not be afraid, for she was set apart for you before the world existed. You will save her, and she will go with you. And I assume that you will have children by her, and they will be like brothers for you. So do not worry.” When Tobiah heard Raphael’s words that she was his kinswoman, and of the lineage of his ancestral house, he loved her deeply, and his heart was truly set on her.

Which of these two passages sound more like traditional Catholic teaching about lust? The Vulgate source clearly shows that lust and impatience in sɛҳuąƖ matters can cause harm. The NAB sources tell you to perform a few superstitious rituals and say a few empty prayers. With the Vulgate, the demon is internal, inside the spiritual battlefield of the person's soul. With the NAB, the demon is something external, outside of the people mysteriously causing death. The Vulgate source contains practical advice regarding continence, fasting, and prayer. The NAB sources create a silly fairy-tale of no practical use.

As for the NAB Bible, in 1991, The Book of Psalms in the NAB was amended to incorporate "gender-neutral language".

The use of "gender-neutral language" is essentially evil.  It is a deliberate imposition of leftist ideology on the Word of God. Even for casual reading, I wouldn't touch any translation that uses it.  This is why I specified using the 2nd Catholic edition of the NIV, because other editions may have "gender-neutral language."  It is obviously an advantage of the D-R that it does not contain any of this nonsense and was made even before anyone could even imagine such foolishness.