As someone else noted, he was too old for the draft, which didn't start until 1969, with men born in 1944 (he was born in 1942).
He could have volunteered, but there's no issue with him doing anything to avoid the draft. By the time they were drafting people, he was in his late 20s and serving his country as an prosecuter.
I was looking on another thread, and somebody said Thompson's problem was that he'd had a lot of extra-marital affairs. I don't know anything about it one way or the other, or whether they were speculating or had seen articles or what. Thompson's senate committee was hot on the illegal campaign contributions to Clinton and Gore (remember the Buddhist nuns testifying that they had been given thousand dollar checks to contribute to Gore at the Buddhist temple?), and suddenly, all that got dropped and they went off on that silly Monica Lewinski trail, and Thompson declined to run for re-election. This was at the time all the FBI files were in the White House.
It could all be coincidence, or there could be something in Thompson's FBI files, and the Clintons also used to have private investigators dig up dirt on people who were causing them problems. If Fred stays silent and out of the race, there's possibly a reason. It could be that he just doesn't want to deal with politics anymore, or the fact that there's something he doesn't want brought up.