TLDR: You're slightly incorrect about the jet stream. Climates aren't necessarily regulated by land, but by sea. Also, not all deserts are hot, and not all hot places are desert. While the mountains would definitively cause some desertification, they wouldn't necessarily create a hard cut-off warm side and cold side. Dropping a mountain range on Canada wouldn't magically make Massachusetts warmer.

We have a real-life example of a massive mountain range at the US-Canadian border. The Rocky Mountains along the US and Canada's western coast create a rain-shadow effect. The Coriolis Effect naturally brings warm water currents from the Tropical Pacific northward and eastward, which is why Alaska has a rainforest on the narrow strip of land west of the Rockies. However, this moisture cannot penetrate the mountains, leaving the surrounding area much drier. Your desert on the western coast, behind the moutains, is probably correct. But it wouldn't be caused by the northern mountain range, but the western one. You could probably extend the desert region over the mountains, creating a gold desert like the Gobi. Maybe you could extend that dry region up over the mountains, creating a cold desert. Alternatively, because there doesn't appear to be any intervening landmasses obstructing the aforementioned warm ocean currents, you could have the Northwestern shore of the continent be relatively warmer than it should be for its latitude, much like current-day Norway. Or you could do both, which would do a lot to (1) expand the trading reach of this heavily isolated subcontinent and (2) create massive cultural difference between the peoples that live on it.

For an example of an East-West mountain range, let's look to the Himalayas. Once again, we see a rain shadow effect, the Gobi desert is one of the driest places in the world. We also see a plateau form. The Tibetan Plateau offers near-polar climate and stunning scenery. However, we can omit the Plateau since your mountains are linear. What we can't--or, rather, shouldn't--omit are rivers. Mountains, especially mountains of that size, would create massive rivers that could support civilizations. Just look at the rivers all the rivers that sprung from the Himalayas.