Need to refrigerate vinaigrette?
Yes, you should refrigerate it.

Once a commercially bottled dressing is exposed to the air it is contaminated. Your homemade dressing is no different, and is likely more contaminated to begin with. The bottom line is that your dressing is food -- for humans, bacteria, and mold. It might not be the most hospitable of environments, but something is bound to find a home there. This process is arrested by refrigeration. Additionally, the garlic and basil will be going through a decay process that would be delayed by refrigeration.

I'm not sure what kind of olive oil you used, but I've never had a vinaigrette turn solid in the refrigerator. If you're in a rush I'd suggest submerging, or running hot water over the bottle you store it in to quickly thaw it. You'll shorten the shelf life, but if you're using it frequently enough (used all in less than a month) that shouldn't matter.