You get downvoting privileges at a certain karma (since about a year ago, 500 points; this is periodically raised as the average user's karma goes up.) Flagging ("this is spam") and the ability to make polls are also based off karma.
Karma has no effect on the rank of your submission or the weight of your votes. However, average per-comment karma has a (mild) effect on the ranking of comments. Additionally, flagging has a severe effect on the ranking of submissions (negates ~10 upvotes, sufficiently-flagged submissions go [dead]). Flagging does not seem to change the ordering of comments, but too many flags make a comment go [dead]. Users that (repeatedly) make comments that get flagged [dead] tend to find themselves hellbanned.
If the average user karma is periodically raised, and downvoting privileges are somewhat based upon this, then over time does it become more difficult to become an influential member?
It's (by definition almost) fairly easy to become an influential member of a group of 3. Much harder in a group of 300,000. This has little to do with how karma etc. works.
Karma has no effect on the rank of your submission or the weight of your votes. However, average per-comment karma has a (mild) effect on the ranking of comments. Additionally, flagging has a severe effect on the ranking of submissions (negates ~10 upvotes, sufficiently-flagged submissions go [dead]). Flagging does not seem to change the ordering of comments, but too many flags make a comment go [dead]. Users that (repeatedly) make comments that get flagged [dead] tend to find themselves hellbanned.