Anything with VP chips on it will change the marginal return on cards, so yes this will influence my strategy.
For instance, say Expedition is out. Early - mid game I hit $3. I could buy a Silver and have a permanent increase in buying power and likely cash density ... or I could get three more cards and boost my next hand. There will be times where silver is clearly better and times when Expedition is clearly better, but with Baths it is more like Expedition + a freebie empty-handed Island. For instance if I am I gambling to hit $6 or $7 when my opponent has a 40% chance of using Militia, 2VP may swing things towards taking the plunge. Other events, like Mission, Pilgrimage, Training, Lost Arts, Tax, etc. all change their efficiency with Baths out.
A particularly nasty option heavily enabled by Baths is to go for a Salt Baths rush strategy. You need to hit $4 per turn and you gain 3 VP, 0 cards per turn. Coppers are helpful so just about anything that gives you net +buy or copper gain can let you end the game by T10 with 23 VP. Sure it can be beaten, but that is a lot of Duchies to grab (if bought singly the other guy needs to start on T3 and never whiff; an engine needs to start on T6 with a singleton and then hit doubles every turn but the last when they can take the final province) and with the VP from Baths that means you need to snag 3 Provinces to compete (and each province you snag kicks the turn counter down by one). Beggar, for instance, would tempt me to open Beggar/Salt. I have pretty good odds that I will not whiff on T3-T4 and once I get those coppers my odds of hitting $3 or less goes down substantially (you will have 6 Baths turns and 8 Salt turns, so you can play Beggar or any other copper gainer 2x in the early game to improve your odds of hitting $4).
More generally, Baths reward early trashing strategies (like say opening double Steward) over early gaining strategies (like opening Iw/Wshop to rapidly build out a Village/Smithy engine). The also change tactical concerns (e.g. top deck a copper with Count instead of gaining a replacement) that impact at the margins or like how Opost will likely become a decent play a shuffle or two sooner as you can always use it as a Monument.
As far as when to "buy" Baths just for the points - well I think early game you should go with "Would I buy Ghall here without anything fancy (e.g. Ironmonger, Expand, Scout, Apothecary)?" and late game it should bought instead of Duchy until the VP run out, you reach the point where you would buy estates, or you have some reason to think you will have more sub-duchy hands than your opponent (e.g. you go BM while he goes engine).