You can book online directly on their website. You used to be able to reserve without providing payment, but that's changed; you now need to pay up front. There are certain circumstances you can change or cancel--check their website for those.
The bookings tend to go pretty quickly, so it's best to get in soon. The catch, of course, is that you have to commit to the scheduling well in advance. So cross your fingers that the cherry blossoms are in bloom and the weather is nice. And be sure to read their cancellation policy in case you need to reschedule or the weather turns nasty.
If you miss out on an advance reservation, all certainly isn't lost. They have a total of over 70 boats, and only 25 are available for reservations. The rest are kept for walk-ups. So it’s still very possible to get a boat on a first-come, first-served basis if you're willing to chance the crowds.
The paddle boats are open 7 days a week from March 13 through the rest of the summer. The first boats go out at 10am and the latest reservations are for 4pm.
The prices are $18/hour for a 2-passenger boat and $30/hour for a 4-person boat. They now also offer 2-passenger swan boats (ie. shaped like a giant swan) that are $30/hour.
Life vests are mandatory, supplied, and included in the price. At least one paddler on each boat has to be at least 16 years old, but kids are welcome. They supply life vests for children over 18 months (or over 25 pounds).
You can find their website here.