The movies we are watching are from the 30s and 40s, so you can assume they will be of their time -- both in a good way and in a bad way. Villains are villainous, ladies need saving, there's gunplay and fisticuffs. The world is pretty white, and if there is someone of color they're likely to be a cook or a porter (although not always!). If you are watching old films (not just westerns) with a kid, you are probably aware of this sort of thing, and how you choose to discuss it and/or mitigate it is your business. I'm not going to bother pointing out the general issues that arise on old films.
Generally, I don't see much that makes me cringe in Autry films, but there are moments here and there and sometimes serious I-want-to-cover-my-kid's-eyes-and-ears moments. The sort of thing where you don't necessarily need to skip the movie, but you might want to be prepared before your 6-year-old makes WTF? face at you. That's the stuff I wish there was a guide for, and that's the stuff I will be mentioning. Because no-one needs to be caught unawares by unexpected blackface.