Currently Online 
I'm going to give my own go at it, but will try to do it sort of "scientifically." Not really, but I'm going to TRY to use some objective factors.
Step 1: Eliminate every franchise below a certain number of sales, as a rough measure of success and popularity. I'm going with 50 million as an arbitrary large number that still leaves us with a few dozen franchises. This doesn't take things like arcade revenue or free-to-play games into account, but what am I going to do?
Step 2: It's hard to determine how you can call a game "influential," so I'm going with a bit of a weird way to measure it. I'll be going to the website TV Tropes and using their articles on "Trope Makers," which would be games/series that created concepts, and "Trope Codifiers," which popularized and/or provided a framework for future works.
Step 3: This one is a bit arbitrary, but I'm going to discount any "franchise" that's basically "one big hit and its spinoffs." I feel that games like Minecraft, for example, would be better reserved for "Top GAME" list instead of a Franchise list.
This already leaves us with just 20 series:
- Borderlands
- Call of Duty
- Dragon Quest
- Elder Scrolls
- Final Fantasy
- Grand Theft Auto
- Halo
- Legend of Zelda
- Mario
- Metal Gear
- Pokemon
- Resident Evil
- Sonic
- Street Fighter
- Super Smash Bros.
- Tekken
- Tetris
- Tomb Raider
- Warcraft
- Worms
From here, it's a matter of comparing these 20 franchises directly against one another. I think most of us could agree that some of these franchises deserve more recognition than others (shoutout to Worms for getting this far). OP mentioned criteria such as modern prevalence and worldwide vs. regional appeal, but you could just as easily have something else in mind.
In no particular order, I'd pick Call of Duty, Final Fantasy, Grand Theft Auto, Legend of Zelda, Mario, Pokemon, Resident Evil, Street Fighter, Tetris, and Warcraft. I'm picking these because each of these series have either been major industry forces for a long time across platforms and regions. With that being said, I wouldn't argue with somebody who put Sonic, Tekken, or Halo on their Top 10.
I don't think I need to justify any particular game in the prior paragraph, but feel free to reply if you want me to elaborate.
