I have a russian tortoise and just got a red eared slider today. Could they pass some pretty bad diseases between eachother? Should I make sure to wash my hands before I handle both? Would it be bad to let them bask together?

Never let them have contact. Yes, you should wash between handleing.

UMDRunner, A Red Ear Slider is an aquatic species and Russians are a terestrial species one is a tortoise and the other a turtle though I am sure you know this. My daughters Red ear Sliders are always in her pond basking in the center of it on rocks. The smaller ones are in an aquarium.
So I am a little confused with the question. How would they bask together? Not that I am suggesting they do (I feel they should not have contact).

I was just thinking if I ever let them outside together in my outdoor enclosure

You should have them in 2 seperate outside enclosures. I watch my three toed box turtle, and a painted turtle closely and always put them on the oposite ends of my backyard, when I clean the painted turtle container.