Do you have a budget in mind?

I personally found 1-year old a tough age to travel with a kid. Younger is easier and older is more easily occupied. I'd focus on short flight times. Is Mexico not an option?

Budget is somewhat flexible. I think you're right about a one-year-old. Where in Mexico would you suggest?

The Bahamas will be too cool in March.

Turks & Cacios would be quite with a fab beach and condo type units to choose from. The Sands is very family friendly.

In Mexico look at the boutique hotels and villas on and around 5th Avenue in Playa del Carmen.

http://www.locogringo.com/dynamic/lodging.cfm?area=Playa%20del%20Carmen&type=Condominium

http://www.locogringo.com/dynamic/lodging.cfm?area=Playa%20del%20Carmen&type=Villa

Thank you! (For clarification, we'd be traveling this month

I vote for Virgin Gorda.

Depending on our group size, we rent a few different villas in Nail Bay.

I think this one would be great for you as you could have the main house for privacy and the nanny could stay in the separate house. It's where I put my in-laws

http://www.vrbo.com/357635

Take a look also at any villas on Mahoe Bay on Virgin Gorda- flat, calm, protected bay for great beach play for your child but also an easy place for you to float and enjoy. A 5-10 minute ride to restaurants and grocery in town. Or since you are traveling so far, take a look also at St. John. USVI- either way, check out flights into STT.

Maui is a great idea, although it is built up and touristy. Mid March is a good time of year to visit, it has many condos and great beaches, personally I love it around Napili Bay, it is popular with families, unspoilt with lots of low rise condos. For the Caribbean, Barbados has beach villas and condos, the west coast beaches are calm and great for families but might be hard to reach from southern California. There are many islands that could fit the bill eg Anguilla but the flights may be a problem.

The Bahamas would still be too cool in December. That destination is better mid May to September weather wise.

You all rock! For Virgin Gorda, are there direct flights from LA I wonder?

I wouldn't want to fly all the way from LA to the Caribbean with a 1-year-old. But maybe your baby is an amazing traveler! If not, I'd stick with Maui or try Cabo. Good luck!