I'd treat it just like a .30-06 when it comes to choosing a scope. In other words, any quality 2-7, 3-9, 4-12 or similar range of scope. I'd want a reticle that looks like crosshairs rather than a street map of New York City and an objective lens no bigger than 40mm. A weight of...as little as possible; scopes of a pound or more are definitely out of the question, and several ounces less would be preferable and easily achieved. No parallax adjustment needed or desired; learn to get into shooting position quickly and consistently with your eye centered behind the eyepiece, and you don't need one. Decent eye relief is a must; a .375 is a hunting rifle, and might be shot from some less-than-ideal positions in the field, so I want that scope as far forward as I can comfortably use. I want an eyepiece that doesn't rotate when the power selector is changed, because I want flip-open lens covers and they don't go well with rotating eyepieces. I want low, capped turrets that are easily finger-adjustable when the covers are off.
And finally...I want two of these scopes, fitted into quality QD rings and both sighted-in and ready to go. Ideally, one would be in the higher and the other in the lower power range; either one might be the "primary" scope on any given hunt depending upon conditions.