I think it's impractical for Board members to sign checks, they are simply
not as available as professional staff.  Of course, use your head -- the
person who keeps the books should not have check signing authority (no matter
how much you trust them!).
Is your staff very small?  If that's the case I can see that the Board would
have to take more of a physically active role rather than a conceptually
active one.

The ED is responsible (legally) but not LIABLE for the actions of the staff,
with complete oversight he/she should be at least ONE of the check signers.