What is the effect of the "1 for last card" score on tactics in cribbage?
I believe you are misreading the rules. As this set points out, the one for last card is just an alias for the minimum go guaranteed to Dealer for playing second.

Update:
Tactically, the guaranteed go for Dealer really only affects end-play as both players approach 121 points. at other times in the game, worrying about a single point is just not important enough.

For example: Consider a situation where Dealer is at 106 and Non-Dealer at 112 with 7 points in hand. Non-Dealer would like to peg at least two points to guarantee the win with first count, but must be wary of sequences that allow Dealer to peg a 2+12, followed by the guaranteed go. Non-Dealer should not lead with a low pair hoping for a triple, as the 6 is insufficient compared to Dealer's possible 2+12. Better to try to get a 15 and give up a possible 6 to Dealer, as that is enough with first count.