Requiring credit card input would bar a lot of young players who don't want to bother their parents with this sort of thing or even adult players who don't want to give out their CC info to play a game, so a less restrictive alternative is to have a good moderation system in place and making an effort to keep minors safe (having an easily accessible system to report abuses and responding promptly to them).
This paired with a solid (and actively enforced) Terms of Service would demonstrate enough good faith on the part of the site operators to protect you from liability in most cases. You can also look at the most popular major gaming networks (PSN, Xbox Live, etc.) and follow them.
The other alternative, of course, is to simply limit inter-player interaction. If there are no text/voice chat capabilities, and no unvetted custom avatars, then all you really need to worry about are inappropriate usernames.
Since you do want to charge for your service anyway, then you could employ a two-tiered system:
- During sign-up & credit card entry, parents can specify that they want the account to be restricted to:
- only gameplay interaction, and no chat/messaging with other players
- or, do as Nintendo did, and limit inter-player social interactions to real-life-based social networks. I.e. in order to chat/message with another player, they need to be added to your buddy list by entering a player identification code that you've previously exchanged at school, by phone, email, etc. This way, kids are only interacting with players that they know in real life and already communicate with.
- only being able to participate in games/chat lobbies watched over by a live moderator
- only seeing user-uploaded content that has been viewed & approved by staff
- Otherwise, the player can play and communicate with any other online player and have immediate access to user-uploaded content that hasn't been vetted by staff yet.
In a freemium model, you can also let users play without entering a CC in the child-safe play mode.