Don't mark a run as verified without doing some kind of check first. You can be more lenient towards people you trust but don't just let runs through unchecked.

I generally check videos like so:
     1. Check the start of the video for the difficulty.
	If the mission selection isn't shown, you can see the modifiers at the start of the mission when they call their Sparrow.
	Worst-case scenario, scan through until you find enemies with their level above their head.
	Keep in mind that for anyone doing missions at level 30+, only DLC missions are "Normal" difficulty.
	Strike playlists show their difficulty when selecting them on the Director. You may need to double-check this yourself.
     2. Check the video timestamps for the mission's start and end.
	Make sure the length of the run is roughly what they submitted, but be aware that cutscenes mess with this timing.
     3. Check the end of the video for the game time on the results screen.
	Reject the video if it isn't shown.

And for Strikes/Raids:
     4. Make a decent effort to check how many people participated.
	Example: if four different people were in a strike (drop out -> place filled), the run is invalid, no exceptions.

Steps 1, 2, and 4 can become more lax if you trust the person submitting, but you should always be making sure the video includes anything before the black screen at the start of the mission, and the results screen after.

Don't ever add mod comments to a run when verifying. Comments should only be added if something in the future affects an old run. Example: "MOD: This run was submitted before [rule change]."

If for any reason you aren't sure if a run should be verified/rejected, you can contact me. I'm most active on Twitter, but I also check Twitch and the speedrun.com forums regularly. YouTube has already proven that its messaging service is awful.

Outside of run verification, the forum is easy enough to moderate. Answer questions that people ask (feel free to post "I don't know, I'll ask another mod"), and if you feel the need to delete anything, make sure you tell the user why it was deleted. Do NOT abuse your power if you disagree with a user. Contact another mod if you're having trouble keeping your cool.

If you're willing to put in the effort, welcome to the team.

~ Nacho