I am similar to above. I like to use past sales and get averages of them, if possible.
Beckett is pretty useless. However if it is a rarer card, in the case where another player from the set might be selling, i like to refer to a beckett to determine relative player pricing within a certain set.
So if player B in the same set sold for X and i am making an offer on player A, i like to check the beckett to see what percentage of book value the 2 players are in the beckett and sometimes compare things like that.
Although i have noticed this is not all that consistent anyway. This probably doesn't work that well for current players where "they can be hot or cold" all of a sudden, but for older cards with players that have already established careers, it can be useful.
Where possible i always like to have some evidence or justification for an offer anyway. I think if you can provide something a reasonable person would offer, based on evidence (like in a court room ;p) it is at least a fair offer.
Whether the seller wants that or not, that's another question. But you should not be be scoffed at or questioned for low balling etc.