I wonder if this would be immoral to do, what with all the cult of the body and immodesty inherent in fitness today.

I could personally be clothed modestly in public if I were to exercise in public (which I never do), but if you were to own a gym, you wouldn't be able to impose the public to only dress modestly, and coaching people seems problematic too.

I wonder if you could just put these things on the side and not worry about it, since exercising and coaching people are not inherently evil and what anyone does is their own business.

Based on what Pius XI wrote on coeducation, I certainly think that he would condemn coed gyms:

http://www.papalencyclicals.net/Pius11/P11RAPPR.HTM

68. False also and harmful to Christian education is the so-called method of "coeducation." This too, by many of its supporters, is founded upon naturalism and the denial of original sin...These principles, with due regard to time and place, must, in accordance with Christian prudence, be applied to all schools, particularly in the most delicate and decisive period of formation, that, namely, of adolescence; and in gymnastic exercises and deportment, special care must be had of Christian modesty in young women and girls, which is so gravely impaired by any kind of exhibition in public.

There are gyms only for ladies.

Before I "reverted" to the Catholic Faith (grew up typical, lax Novus Ordo), I was a personal fitness trainer.  It was very problematic in many ways.  

Gyms are one of the least immoral places to own.  People generally don't go to gyms to make fornication dates.  My experience is that people rush to the gym to fit a work out into a busy schedule. People don't intentionally dress immodestly, either.  Perhaps that is the case by traditional catholic standards, but by society's standards, no.  Exercise and a healthy diet is a positive thing.  "Cult of the body" is far removed from a healthy lifestyle.

If you owned a gym, you can set the tone by having a dress code for your employees - no skin tight clothes.  Successful gyms are often segregated by sex, too.  Women often want to work out without the pressure of men around.  Men often have different exercise needs and desires such as free weights and less cardio.

If you want to see an immoral place, try a public school or college.  Shopping malls are also chock full of scandal. 

Trad Gyms definitely need a dress code... and no fat Yids allowed!