It's tasteless because it completely misses the point.
Joseph was kind enough to take Mary in. This was a tradition at the time, however not every guy would do it. This applies today, as well as back then. Yes, this might have been a virgin birth, and Mary might have been the mother of God, and all that. Joseph didn't really know that for sure. Nevertheless, Joseph took Mary in and looked after her. This cartoon makes fun of that commitment.
Further, the comic goes on to make fun of Joseph being unable to "perform" for Mary. The inability to perform under pressure has got to be one of the most well documented sexual problems in new couples. If Mary is a virgin, this can happen. If Mary isn't a virgin, this still can happen. Either way, the comic is really making fun of a very common insecurity.
Further, any reading of the bible, does not make the entire Jesus birth sequence a "fun" thing for either Mary or Joseph. So what is the cartoon trying to say: "here is another way for Jesus birth not to be fun?" Keep in mind, as the bible story goes, shortly King Herod is going to kill a whole bunch of young Jewish children. Being unable to perform in bed is a bigger problem than having someone try to kill your kid? Being unable to perform is worse than a near genocide?
Additionally, the cartoon even fails in its stated intent. The cartoon's purpose is to make fun of the Virgin Birth. Just because God has made love to a girl, does that mean I can't? Nuns are married to God. They occasionally have sex, and get pregnant. I don't see how Joseph's inferiority has anything to do with the Virgin Birth. Surely, Joseph and Mary would have had a full married life, including sex.
Ultimately, you either believe in the virgin birth and a caring God, or you don't. If you believe in the virgin birth and a caring God, then you probably believe that a caring God wouldn't do this to Joseph. If you don't believe in either the virgin birth or a caring God, then you probably believe Joseph got Mary pregnant out of wedlock. In either interpretation, the cartoon looks silly.
In short, this cartoon is tasteless, because it misses the point in so many different ways.