You can't really join the Blades, the most you can do is obey their order to
and thus stay in their good graces, which will allow to recruit new members for Delphine (if you haven't finished with that already) and continue to do whatever remaining quests you can get from Esbern.
If you do that, you will no longer be welcome in High Hrothgar (altho I don't think the GBs do anything to you other than give you the cold shoulder if you go there) and Arngeir will no longer give you the locations of additional words of power, as already noted.
If you are on PC there is a mod that allows you to refuse the Blades' requests mentioned above and convince them to accept that, resulting in their disposition towards you returning to what it was initially and allowing you to do all the Blades related stuff that you might not be able to do otherwise.
Regarding their history as servants of the Empire/Emperor, that's pretty much null and void at this point - they were disbanded if not totally disavowed by the Empire 30 years ago per the terms of the WGC and ceased to have any official function within the Empire from that point on. Given Delphine's character and temperament I'm guessing she harbors little if any lingering sense of loyalty to the people who basically left her and her comrades high and dry after the war, forcing them to go into hiding for their own protection and as far as I can tell turning a blind eye to Thalmor attempts to find and kill them. Esbern doesn't seem like the bitter, hold-a-grudge type but he's probably too wrapped up in his dragon obsession to give it much thought. Both of them seem intent on reforming the Blades as a dragonslaying outfit with little interest in anything else.