No, clothing doesn't transmit STDs. They are only sexually transmitted. They either need direct skin to skin contact or penetration to transmit.
However, if you are sharing underwear, it's possible that fungal infections like jock itch, yeast and the like could be transmitted this way. You shouldn't be sharing underwear. It's fine if it's clean, but since you don't know if it is or not, the practice should be that he just doesn't wear your underwear, to be safe.
If he doesn't have enough of his own, either get him more or do the laundry more often.