I've played with bribing allowed in several groups, and I've found it adds strategic depth and increases liquidity for trades.
Because players in my groups always keep deals, we've actually commoditized protection -- a player often sells the right to be left-alone the next time he moves the robber, a right we call a "Not-Hit."
This isn't worth a full resource card, so it's usually used to sweeten some borderline deal. For example, if I'm selling my sheep to someone who has a rock, a wheat, and a brick, I might request he give me a single Not-Hit in addition to the brick, lest I help make a Knight that will harm me later. Or I might simply offer a Not-Hit for a single use of someone's 3-1 port, which ordinarily isn't worth a whole card.
Often players are willing to accept a Not-Hit as a bribe, at least from 1-2 opponents, so fewer real resource cards trade hands than you might expect. We also sometimes perform coerced trades -- I agree not to rob you, but you have to trade your valuable brick for my worthless stick.
Late-game, it's so valuable to rob the winning player that bribes simply don't happen. It's important enough to slow down the winner that the robber isn't a credible threat to the losing players.
My groups play with resource-cards face-up*, which strengthens the robber as you know whom you're going to target, but despite that we find bribes make Development cards much less variable in value. Monopoly, Road Building, and Year of Plenty all just-about pay for themselves (if not more-so), but without bribes, Knights aren't so great, despite letting you chase Largest Army.
In our games, playing a Knight usually gets you 0-2 crummy spare resource cards and 1-2 Not-Hits, plus 1 random resource card. In a certain sense, it's a hybrid between a Year of Plenty and a VP card (as it chases Largest Army), making it about even with the rest of the deck.
The main harm in having bribes is that rolling a 7 is luckier. However, this isn't much worse than the standard variability in the game -- some rolls generally help certain players much more than others.
*The robber takes a random card from the face-up hand.