The Hamilton equations are
$$ \dot{q}_k=\frac{\partial H}{\partial p_k}~~~~-\dot{p}_k=\frac{\partial H}{\partial q_k}~~~~~-\frac{\partial L}{\partial t}=\frac{\partial H}{\partial t}.$$
Does the minus sign of the momenta equation, have a physical meaning?
Consider the Hamiltonian for a single particle in some potential $$H(p,q) = \frac{p^2}{2m} + V(q)$$ where the first term is the kinetic energy and the second term is the potential energy. In this case, we have $$\frac{\partial H}{\partial q} = \frac{\partial V}{\partial q}$$ On the other hand, the momentum is $$p = mv$$ where $m$ is the mass and $v$ is the velocity, so $$\dot{p} = \frac{d}{dt}{m v} = m\dot{v} = m a$$ where $a$ is the acceleration. In this case, the Hamilton equation $$-\dot{p} = \frac{\partial H}{\partial q}$$ becomes $$-m a = \frac{\partial V}{\partial q}$$ If we now use Newton's second law $$F = ma,$$ we get $$F = - \frac{\partial V}{\partial q}$$ If we didn't have the minus sign in the Hamilton equation, we wouldn't have the minus sign here. It tells you that the Force associated with a potential always acts in the direction which makes it smaller. (If $\partial V/\partial q$ is positive, that means $V$ gets larger when $q$ becomes larger, so $F$ acts in the opposite direction.)
This is why things fall down. Because the gravitational potential gets lower in that direction. If the minus sign wasn't there, everything would fall up instead of down. So it's pretty important in keeping the universe together.
More broadly speaking, the asymmetric minus sign in Hamilton's equations is intimately related to the antisymmetry of the symplectic/Poisson structure, which in turn has several consequences, e.g.
Liouville's theorem, i.e. that Hamiltonian vector fields are divergence-free.
An autonomous Hamiltonian is an integral of motion.
For more informations, see also this, this, this & this related Phys.SE posts.