How should coax cable enter the house in a harsh climate?
I don't think it matters all that much. I can't find a minimum operating temp for coax, but most wires tend to list -20C as their minimum, and that's mostly for bending reasons. So if it got hit in the middle of winter, maybe it might get damaged. Maybe...

As far as weatherproof... What you have in your picture is the grounding block. I have plenty of coax connections exposed to the elements. I've not seen them experience anything bad from it. More modern installs tend to put this inside a plastic box but I had a satellite dish with a block like that and it was fully exposed as well.

If I were you, however, I would check the wire going into the house. My bet is it's RG59. If you're looking to improve this you could


Replace the RG59 with RG6 (quad shield if you can get it). You'll get a better signal, which is important for things like cable modems.
Put it in conduit. PVC electrical conduit is cheap and if you can limit exposure to the elements it's worth the effort.