Lecce for 3-4 weeks with kids... is it worth it?

So I guess you were not planning on renting a car? It's about a 20 minute drive to the closest beach, not sure what public transport would be like in May. It is not the easiest public transport region in Italy, but I'm sure it's not impossible. It will require research though.
I would do at least two bases, Lecce would be great for lots to do, but I would want to be at the seaside, too for a change of pace--Gallipoli, Otranto, Monopoli might be nice.
Parts of Lecce can have a slightly dodgy feel, but I never felt concerned as a female walking around in the historic district.

I don't know this area. But here's the Lecce city bus system, for a start: https://www.sgmlecce.it/servizi/trasporto-pubblico/mappa-linee.

Lecce is served by Trenitalia.com and rail maps show a fair number of lines that are likely to offer slower, regional train service around the heel of the boot, with a number of connections. In other cases, a search engine like www.rome2rio.com can help to identify bus or private-train alternatives. For instance, some separate railways north of Lecce include Gargano Railways (from San Severo), Southeast Railways (Bari–Alberobello–Taranto tickets cost about $10), and Bari–Matera trains ($5 tickets) (see links on the sidebar at https://www.ricksteves.com/travel-tips/transportation/trains/italy-rail-passes).

Lonely Planet has a Southern Italy guidebook and you can buy single chapters online if you don't need the whole thing.

I spent several nights in Lecce in 2015, taking advantage of the van trips offered by the tourist office to get to some of the picturesque towns/villages in Puglia. For that reason I didn't need to depend on public transportation. I think as long as you're trying to get to just one place a day, you'll be fine. It's when you're trying to string together a whole series of 2-hour stops by train and public bus that things tend to break down.

I took a walking tour offered by the tourist office and found it very worthwhile.

I was comfortable walking around in Lecce by myself; I never felt nervous at all. I agree that a month seems a fairly long time in a place of that size, especially without a car, and I would prefer to split that time between two cities, but that's just the way I am.

This is the train fir the area, but it seems you can now book many stops on the regular Trenitalia site. Info here.

https://www.fsitaliane.it/content/fsitaliane/en/Media/press-releases-and-news/2018/11/20/winter-timetable-2018-2019-.html

Looked for the new FS timetable but was unable to find it.

I am curious why you selected Lecce. It is lovely.enough but I can think of many more places I would spend a month as a family with more to do and better connections. Have you been to Italy before?

Yes, we lived in Rome for 4 months and have traveled extensively throughout northern Italy. We specifically wanted to experience Puglia since we’ve heard amazing things about the ‘heel’. What other locations would you recommend instead?

I would not take a 4-year-old & 1-year-old to Lecce. When you get that far south in Italy the tourist infrastructure is just not there. Public transportation is not easy. We had to go to three different ticket offices to find the one selling train tickets to Gallipoli. I can’t imagine doing that with 2 small children. And the beaches I saw near Lecce are not the gorgeous white sand you find further north along the Adriatic.

My first choice for you would be Senigallia, further north, in Le Marche. This is a lovely small community bordering miles of flat sandy beaches. The sea is very shallow for a long ways out and there are no big waves, so it’s perfect for children. There are hotels along the beach with enclosed playgrounds on the sand. I don’t know if these are open in May, or if any of them are apartments , but they would be ideal. Senigallia Is a very walkable place, where you could get by without a car. The town has daily markets, a fort to explore, and even nearby Caverns of Frasassi, which may be fun for your 4-year-old. Senigallia is well -connected by train & bus.

Another option would be to stay at a masseria, which is similar to an agriturismo in the rest of Italy. We stayed at one not far from Bari in Puglia. It had a shallow swimming pool, goats and other animals, and lots of fruit trees. You’d pretty much need a car, but the driving is easy. You could fly into Bari or Brindisi & rent your car. Then you could take little day trips to the trulli villages and to the sea — there’s a marine preserve along the coast. On a masseria the kids could run and explore and yet you’d be close enough to Monopoli and other cities for you to enjoy. The masseria we stayed at had a communal kitchen , well stocked, so you could feed the children dinners at home.

Puglia is lovely, but not easy like the north. If you had a car, it would be manageable with children. By public transportation I think you will be stressed. How about somewhere in Campania? Still not as easy as the north but more infrastructure for visitors. Vico Equense (small) might suit you, or even Salerno as there is a high speed trainline there.

I like Charlene’s idea of Senigallia. I have not been there but we did stay in Le Marche right on the sea and it is very tranquil and reasonably well-connected by bus and train. Also very affordable. Perhaps you can base in Senigallia and take a trip to Lecce to check it out for “next time.”

Renting a car will allow you the freedom you need in Puglia. Summer 2019 will be our third trip to Puglia. Three-four weeks is quite a long time, however, if you truly want to settle in and relax without worry of time, you could enjoy being a local. Distances are close in Puglia. I love Lecce with its historic center (I found no safety concerns), but with children the countryside and coastal towns will delight, especially if the family is adventuresome. If you want to locate in the country the area near Alberobello/Locorotondo/Ostuni/Martina Franca might be your fit. I was enchanted by Otranto and told myself on my next visit I would settle in for a few day there. Also Gallipoli, Polignano a Mare (beautiful), and up on the Gargano is the seaside town of Vieste (gorgeous). We look forward to adding Taranto to our list for history sake. Again, you will be limited without a car, but splurging for car will give you the freedom your family will appreciate. Once you have tried driving in the south you will find no issues. If you decide to rent a car do so in your planning stages while still in the U.S. We use Avis or Hertz mostly.