When I visited Monterey before moving here, I went to a coffee shop in Pacific Grove called The Works:
I think PG Juice and Java has free internet, too:
Technically, it's Pacific Grove and not Monterey, but it's just a few blocks from the border between the towns.
In Monterey, East Village has wifi. Everyone I know loves it, but I've never been, so I can't personally vouch for it:
Plumes is smack in the middle of Old Town. Great coffee, free wifi:
If you prefer beer with your wifi, try Ol' Factory Cafe in Sand City: