I'm looking at buying my first boat and not sure how much I have to put down... Most sites say it depends on the boat value but then don't give examples.

The boat I want is a roughly $60K Center Console (New). Would I be good just putting 10% down?

Sure but, it is what ever you can 'comfortably afford'. Most, myself included, don't want a boat payment at all or, for as short a time as you again, can afford.
But, if your budget it 'tight'....then long time financing is great but, will cost you interest money.
Really up to you.....but to answer your question: Yes, 10% should be fine
JMO

God Bless

Thanks Lee and I completely agree with you. The payments aren't an issue. The only reason I don't want to put a lot down is that last year the house flooded and we had to replace all hard wood floors, water heater, paint, etc and went through a huge chunk of savings and have been paying $1200 a month on our remodeling loan to make sure we pay it off before the zero interest period runs out. $6k is just what is be comfortable with right now to still leave our rebuilt savings in tact.

Having a good down payment is always a good idea. SOME lenders have different rates based on the percentage of equity in the loan. If they have lower risk SOMETIMES the rate gets adjusted. Having 10% is normally a good place to start.
Ken