Your browser, which is presumably unpatched in years and never particularly hardened* in the first place, could leak info (you might not want your history splashed around the place, indeed even cookies could be a problem for some), your browser could hose up the SSL connection and leave the data able to be decrypted by other parties (to say nothing of it possibly being old enough to have to use old protocols), if it matters I doubt the browser has a blacklist of bad sites (though many would probably want to go for DNS level there).
*technically the awful javascript performance it has is because of security measures making more dynamic things harder.
Possible to get malware on a 3ds via browser (here kid visit this link on the browser)? I don't think we saw exploits particularly using the browser but I can't rule one out somewhere (especially if it uses various image libraries that do get compromises), and that in turn could be used to do something. I doubt anybody would make such a thing these days (someone that skilled could make a lot more via other means legit and otherwise) though.