Really, the most important thing is to have the ability to deal with monsters that are in the way without having significant delays. Whether that is being able to effectively sneak past things or just kill them outright.
If you're trying to jump around to locations to pick up clues to seal, you want to make sure you can get to locations and not have (sealing critical) people sit in the streets or stay in the same location because they can't deal with the baddie that is sitting in the streets. Similarly, it is frequently worthwhile to have a monster beater rush a gate (and get sucked through) just to clear the way for someone with the ability to seal.
Also note, that it's important to make sure there's someone in Arkham that can deal with monsters in a pinch should the regular monster beater be unavailable (healing, in an OW, etc.).
Pretty much everyone can gather clues equally well, and fully healed people with a health of 5/5 are more likely to survive the OW trip (although it's important to remember that they may have to fight something during their trip, so you don't want them completely defenseless), and there really aren't many people that can't make either the Fight or Lore check to close.
Other than that though, it's always nice having someone that starts with some money to pass around if healing cash is needed in a pinch. Having at least one Magical and Physical weapon to deal with things is nice (Although depending on the expansion, you may want to tote around spells).
All of this can be greatly affected by with AO that is in play. If Shub is on the board, you'll need really good monster fighters to not have the game slow to a crawl.