Well, I wear tight jeans (Wranglers, available for about $15 at Wal-Mart) and 'tighty whities', although they actually come in colors now. Apart from a tendency to sing soprano, I've noticed no problems...
Joking aside, I think a couple of things are important. No - get your mind out of the gutter - not THOSE couple of things. I mean about riding!
1 - If you're riding on your balls, you're either jumping or doing something very wrong. And I honestly think you are doing something wrong if jumping causes contact, but I don't jump so I won't say. Or maybe you need to wear tight jeans. If I'm in an English or Aussie saddle, I'd have to lean WAY over to get contact between the family jewels and the saddle.
Realistically, how many guys can't sit in a chair because their testicles get slammed on the chair when they sit down? Your thighs are not vertical when you sit in the saddle. You are not straddling a 2x4!
The picture below is of an old-time cowboy. They rode with a straighter leg than most modern riders.
Even there, his thigh is forward of vertical, bearing weight and preventing his balls from contacting the saddle. Or consider the ancient Greeks, riding naked:
If riding a horse required contact between the balls and the horse, men would never have tamed horses.
2 - Western saddles can be more of a problem. The typical western saddle now-a-days is marketed to women. They look more like a torture trap for men:
This is the "Julie Goodnight Circle Y Teton Trail Flex2 Saddle" - maybe it works for Julie, but Bob has no desire to ride in it!
A better design for men is the old style Wade ranch:
Most English and Aussie saddles avoid the stupidity of the 'saddle for women who hate men'. This is the one I mostly use:
Like most English saddles, it allows a man to be a man.
Please remember that men rode horses for thousands of years before it became common for women to ride. It says a lot about how depressingly female dominated riding has become that folks wonder if men can actually ride a horse without pain.