"A base" is any one of the three Roxxon labs (that are unlocked after story mission 9 "Corporate Espionage") or three Underground hideouts (unlocked after story mission 10 "Underground Undercover".) These are the same six locations needed to earn the Under Their Noses and Underground Undone trophies, respectively.
The wording of the trophy also misleads a bit. By "without being detected" it actually means "without triggering open combat." I decided to do this trophy on the West Side Roxxon reactor because it just so happened to be the one I was nearest to after unlocking them. After taking out about 1/4 of the enemies there was radio chatter about someone not checking in, whereafter every enemy in the base started yelling about "finding Spiderman" and kept finding all their knocked out/webbed up friends. Many of them had white question mark indicators, and some started filling up yellow (being detected) but I got the trophy in the end anyway because open combat was never triggered. Using environmental hazards does not void the trophy.
Camouflage is very useful in general, which makes repositioning for takedowns a breeze and is also a helpful failsafe for avoiding detection. I also used the Eyes On Target visor mod that draws blue lines to show you who is looking at your target, so you can work inwards from the outermost enemies with ease. I also elected to use the Optic Triangulation visor mod because it permanently marks all enemies in the area after using
once. Ghost Strike and Eyes In The Back Of My Head are also very useful, since the first makes camouflaged takedowns silent and the second automatically activates camo right before open combat would be triggered.