Lindsay's answer is pretty spot on, and very similar to how I would approach things.
If I notice the stay is "pending" the day before the booking, I'll send a message reminding them to book and giving them clear instructions on how to do so:
"Hey (client). I'm really looking forward to having (pet) over today! I've confirmed the booking on my end. When you get a chance, please log into the app or website and confirm it. This will make sure that (pet) is all setup by (drop-off/pick-up time). If you're having any trouble, let me know and I can walk you through it".
If the day of the stay arrives:
"Hi there! I'm really hoping that you're still planning on bringing (pet) over today. I'm really looking forward to getting to know them and having a lot of fun over their stay. I noticed that the booking still needs to be confirmed. Please let me know if plans have changed or feel free to give me a call at (Rover #) if I can help you."
I always give my clients up to two hours before the booking is supposed to start. Then I give them a call. Life can get busy. And not everyone has their messages sent to their phone numbers. If they don't answer, I leave a voicemail:
"This is Shane from Rover. I just wanted to call and check on our plans for (pet)'s stay starting in a couple hours. I checked with Rover and it looks like the booking is just waiting on you to confirm. If you're still planning to bring (pet) over, please let me know. You can message me through the Rover platform or call me back on this number"
If I don't hear back within the hour, I cancel the booking on my end and open up the spot for another last minute booking (should I get a request). If a client is unable to respond after two messages and a phone call, but then decides to show up anyway, they are usually the client that is 2 hours late for pick up, changing plans last minute, etc.
There have been times where the client shows up and the booking still isn't confirmed. If I can tell it was because they didn't know how to use the app or weren't aware that they needed to confirm, I'll generally show them how to do it and complete the booking for them on their phone.