I want to edit a negative effect from a mod but the script that is controlling the effect is inside a BSA file. Which program(s) do I need to open the BSA file and edit the script that is inside it?

Thanks.

Scripts are compiled, so they cannot easily be edited even from inside the BSA.
At a minimum, you'll need to hope that the mod author has included the source file.
Then you'll need a program such as BSA Extractor to extract the BSA, and you'll need the papyrus compiler included with the Creation Kit to compile the script after you modify it ...

No source file included. The mod only has the esp, bsa, and bsl files (what is a bsl file anyway?). I tried contacting the mod author but it's been 4 days now and no reply from him. I'm SOL then?

The source files may be in the BSA; they often are.

You need to extract the BSA file, they're like ZIP files just in gamesas's own format. Use the http://skyrim.nexusmods.com/downloads/file.php?id=1395 to unpack them.

The BSA contains all the mod's data files. If the modder was nice he also included the script sources.