AC and exploration mode
It will down-level you if you’re above the dungeon’s level, but it won’t up-level you.
Soloing at 60 with no prior dungeon experience would not be advisable. You’ll want to learn the paths with a party before trying them on your own, and be properly geared in order to deal as much damage as needed (e.g. lvl 80 exotics, minimum).
Also, most LFG groups aren’t fans of <80s, even with the downscaling. A level 60 won’t be geared as well or have all of their traits, and some groups will kick you on sight.
I’d suggest putting up an LFG with a message like “first time, low-levels welcome, teacher wanted” or similar.
If you really want to get into running dungeons, the best way is to join a good guild. If you’re NA I can send you an invite to mine if you’re interested — we’re pretty laid back, enjoy dungeons, and don’t mind teaching or taking lower level players :-)