Sarahjane, welcome to TP. I'm sorry it's so stressful with your mother.
I'm another one who has experienced the rude and loud and socially inappropriate and cringe-worthy comments in public, both from my grandmother (who used very racist language) and, currently, from my mother, both of whom had/have Alzheimer's type dementia. My mother currently has a delusion/fixation about a man at her care home whom, she says, drops his trousers and, um, touches himself inappropriately. Her very loud descriptions, complete with hand gestures, of the, er, inappropriate touching make me really want to melt into the floor. I suppose I should be grateful she's not doing sound effects as well.
So you're definitely not the only one out there who has experienced this kind of behaviour. Not that it makes it better for you while it's happening, but I personally find it comforting to know I'm not alone or not the only one going through something.
I don't have great advice for you. With my mother, I've found the best strategy is to make a concerned/sympathetic/surprised comment, such as, "he does? Oh, my? Really?" and then distract her/change the subject as quickly as possible, before she can get stuck into her loop on it. The best distraction for my mother is food (coffee and/or biscuits and/or ice cream), which of course we can't always offer, but that's the sure-fire distraction that tends to flip her mood to positive most of the time. And sometimes if I interrupt her too quickly, she doggedly returns to the subject. It's definitely trial-and-error.
I have seen/heard from others, here on TP and in my support groups, that some have success with politely but firmly telling the PWD (person with dementia) that they are not discussing that, not listening to that sort of language, something along those lines, so that's another tactic you could try if you think it appropriate.
If you're not able to distract your mother and redirect the conversation, you can try removing yourself physically when it happens--just stand up and announce you need to use the toilet, and off you go, or suddenly take an "urgent" call on your mobile, or whatever gets you out of the room/area for a moment.
I agree with others who have seen, with dementia, both social skills and "filters" erode as the disease progresses. I imagine this is where a lot of the inappropriate stuff comes from, but don't really know.
When I'm in public with my mother and she is behaving in whatever strange way, I often am able, without my mother noticing, to tip off the clerk or waitress or whomever. Usually all I have to say is, thank you for being patient with my mother, she has Alzheimer's, and I get a flash of sympathy and understanding. Sometimes I don't have to say anything at all. I also have some business-type cards I got from a support group or workshop that say something similar, that I can slip to someone if I need to. I wonder if any of those strategies would help you?
Also as Canadian Joanne points out, self-reflection also somehow becomes impossible. My mother has been slender all of her life, but has recently put on quite a bit of weight. Of course she still thinks of herself as slender, and claims that the laundry is "shrinking" her clothes. Well, no, Mum, actually your trousers don't fit because you're now at least a size larger than you used to be!
Oh, and to get back the OP, I also wanted to say, DO NOT let this behaviour stop you from taking her to day care. They should be more than able to take this in their stride and have probably heard and seen much worse.
I'm sorry this is all so distressing for you.