Sestri Levante or Bonassola or Levanto or…

All three are decent choices, but Levanto is the better town in my opinion. It's close by, has good restaurants, good train service, and is the only one that is on the main ferry line. Sestri Levante is a bit far away and pretty modern. Bonassola is also a good choice, but it's fairly small with not many restaurants.

Consider staying in a town that is connected by ferry--that may be a preferable to arrive and depart:
https://www.apathtolunch.com/2014/06/complete-cinque-terre-portovenere-la.html

I'm going to make an alternative suggestion that you do consider staying in one of the towns. You have said you don't want to stay because of the crowds and steep hills, but if you daytrip in then you are going to be in the worst of the crowds; AND it is busiest in the lower part of the towns. The only way to escape them is to walk up. And even the trains or boats will be busy mid-day.

If you truly want to see Cinque Terre, then it is at it's nicest in the morning before the groups arrive and after they leave. We stayed in Manarola in October at a hotel that was only a very short walk up. We got out as early as we could, then rested in our room at the busiest part of the day, then headed out again after. I don't know if June would be busier later, but if it is, it will be even less appealing to day trip in.

There was a post a few months ago asking for quiet accommodations, I have copied here:
https://community.ricksteves.com/travel-forum/italy/quiet-places-to-stay-in-cinque-terre

I sort of echo Lisa's reply. We stayed in Bonassola one year, and the trains were just unbelievably packed for daytripping to other towns. We wanted to go to Portovenere, which meant passing through all the Cinque Terre towns. It was horrible.

hanks so much, all! First of all, Janet, I live in Boise as well! My name is Marcia. Glad to buy you lunch if you have time to chat about the area!

Lisa, I’d actually love to stay in one of the towns, but my mom can’t walk hills, and is very nervous about going down steps. She’s 85. But I am becoming quite concerned about the cruise ship mobs and the crowded trains. Maybe I should just take a day trip up there from Lucca, and she and I should go someplace else after Lucca. Sad that it’s so crowded during the day.

Mike and valadelphia, I appreciate the suggestion of staying in Levanto or another town on the boat route!

I would vote for Levanto - I've stayed there several times while visiting the Cinque Terre, and it has always been a practical choice for me. It's not like the five villages - not a "postcard" town from the water, but otherwise it's a nice town. I will point out that the train station is up a bit of an incline from the town, and I would want to stay down by the town center not right next to the station (there may be some lodging options near the station for drivers, but I don't think there are any right AT the station).

Sestri Levante has a nice touristy little town center by the water, but that's also not that close to the station. There may be some lodging right at that station, though the area right around the station isn't especially charming.

There is some boat service from Levanto to the villages but not every boat services Levanto as I recall.

But getting on and off boats may be a problem for your mom too. Sending you a PM.

Late to the party so you may have already figured this out, but we are in Santa Margherita Ligure right now and it is very flat. Way less crowded and expensive than CT which is perfect. They do have a ferry that will take you to CT (in April it’s only on Sundays but I think this summer it will be more frequent). My favorite Riviera town has actually been Camogli right next store, although it’s a hill to get down from the train so maybe not good for your mom.

We are staying at Hotel Jolanda which is one that RS recs, but you should be aware there are 5 steps at entrance (elevator is after that) so it’s been a pain with the stroller. So make sure to ask before booking!