Having a lot of Mario experience, i can tell you that the only possible way to beat that guy is to avoid going on plain stages/stages with limited recovery options in general (FD, SV, GHZ). You absolutely need the possibility of being able to recover safely into a top platform without getting caped to your death trying to sweetspot or just recovering in general. It's worth noting, however, that with the proper momentum, Down B is a very safe way to recover, as you can grab the ledge backwards with it and be safe from caping.
It's true, Mario combos Bowser hard and for a few specific %s he can even chaingrab for a decent amount of time. Spacing out and crawling/PSing his fireballs to keep him from approaching and being a threat are things you must practice in this MU.
This thread is about all Bowser MUs though, so:
Fox absolutely wrecks Bowser. His worst MU in the entire cast, as long as the Fox plays properly and camps/baits you out well.
Sheik is rather similar, specially with how easy her grabs and tilts chain together and make your life miserable. Fair is an awful awful move and will get you killed very early on.
Link can be very painful, but the MU is eased somewhat if you can PS consistently. Even then, a good Link will learn to not throw projectiles at you but rather use them to block options and limit movement, and he can easily keep you in a combo all day with Grabs/Tilts/Aerials if you aren't careful.
The 4 MUs i mentioned above are the only ones i genuinely feel are very bad for Bowser. Some others like CF or Wolf are also hard but slighty more manageable, i feel.
When it comes to easy/positive MUs, I would say some stuff like GnW, DK or DDD (Bowser wrecks Bowser, just like Bowser wrecks almost any other big, heavy character.). Even Falco, since his speed is nowhere near as good as Fox's and one of his main approach methods are fairly easy to deal with when using crawl armor or PSing.
Btw, i've mentioned it many times but every Bowser player that wants to get better should master PSing, it's actually a much easier mechanic to perform in PM and it helps you in a lot of MUs that would otherwise be very painful because of projectiles.