Crumb on Parchment- open weekdays and close early but as good as it gets for kids.
Michael's Genuine-outside area or second room work fine. Sunday brunch is great w/ kids.
Sustain- inside or outside work. Their French toast sticks on the Brunch menu are huge hit
Sra. Martinez has outside seating out front which works. Inside can work too depending on time.
Naoe- kids dig the uni... **ok this one is a joke
Sugarcane Raw Bar
Chef Philip Ho- very crowded and noisy during weekend dim sum service. Tons of young kids at tables.
Sakaya Kitchen- casual counter service w/ kick butt food.
Makoto- Outside seating at Bal Harbour w/ great sushi etc
Area 31- outside seating
michaels genuine outside for sure, so good and nice place for baby to look around. sardinia outside or in. lincoln road good pizzas at spris or sosta. books and books, shake shack on lincoln road. i would not do pubbelly with my kids, restaurant is small and many tables are high
There are very few places in Miami that are so stuffy / quiet / fancy that you'd be out of place with a 1-yr old. I probably wouldn't go to DB Bistro Moderne or Palme d'Or, probably wouldn't go to a place like Casa Tua or Cecconi's, but most places it will not be a big deal.
Outside seating at Michael's Genuine is particularly amenable. There's also outdoor seating at Sugarcane Raw Bar Grill and Sustain, and both are noisier places where a little child-related commotion will go largely unnoticed inside too. Red Light is very casual with very good food, only drawback is that service can sometimes be slow. If you go further north, Hiro's Yakko-San always seems to have a screaming baby there every time we've been - maybe once it could be your screaming baby!
On the beach, Pubbelly is a little small and tight, which can be tricky with a high chair and/or having to take a little walk for a crying fit, but Barceloneta is roomier (haven't been to Pubbelly Sushi yet). Yardbird also practically looked like a day-care center when we were there earlier last weekend (though the food has been hit and miss for me). Altamare is a good call too.
Altamare is delicious and very kid friendly. There is indoor as well as outdoor seating and they are one of the few nice restaurants I have found that has a changing table. Altamare doesn't get super crowded, so I don't think it would matter what time of night you dine.
I have recently seen kids at Pubbelly Sushi - which is a new, hip sushi spot. Since Pubbelly Sushi is kid-friendly, I imagine Pubbelly would be too.
I also agree with Pilches, most places will be accommodating, particularly early on in the evening.
I would do barceloneta instead of pubbelly and pubbelly sushi as barceloneta has the best outdoor seating. The food is all about as good as each other at these restaruants. Pubbelly sushi's outdoor seating would be a second choice of the three. Pubbelly's outdoor seating is not suitable for babies.
Altamare is another good choice.