Our daughter was born in September but didn't start sailing until she
was about 8 months old for a 2 week trip. We did live on our 30' boat
though .

When she was 8 months old she wasn't walking so she spent a lot of the
time underway when in a car seat under the dodger to keep her from the
sun. She loved dinghy rides and dragging her hand in the water. A 0-4
month old doesn't need much to keep her amused. Keep her hydrated, out
of the sun, and comfortable and they will be fine.

People that have their babies wear a lifejacket in a car seat sound a
bit much though! If the baby is strapped in the car seat, and the car
seat is attached to the boat, why bother?

Suggestions: have the boat set up to be easily single handed, including
an autopilot if at all possible. Make meals in advance or keep them
very simple when underway. Be conservative about your sailing
destinations & the weather. Allow lots of play time before & after the
sailing part of the day.

At age 8 months she was easier to handle than the following summer, when
we moved, had a smaller (24' San Juan) boat. That summer she wore a PFD
in the cockpit (a tether was not a good idea for her active little
self). This 1-1/2 - 2-1/2 age range was more challenging; as somebody
said lots of activities but not much brains for safety.

New "boat specific" toys & activities when she/he gets older are great
(bubbles, pouring water in the cockpit, pretend fishing rods).

The next year we got a bigger 40' catamaran. She is pretty happy with
her next size up lifejacket. She has to wear it out of the cockpit on
deck. Since our boat doesn't heel we are less worried about losing her,
though one day we'll get lifelines (seriously). At this age she loves
our trampolines and her real fishing rod and going with dad to set the
crab trap.

She has fallen out of a dinghy while rowing with cousins at age 2-3/4.
Wearing her PFD she starting swimming for the big boat. Her only worry
was her hat had covered her eyes and couldn't see too well where she was
going.