In polar molecules specifically, consider the structure of the atoms making up the molecule. Using H2O as an example, as PhysicsDave said, the electrons of the hydrogen atoms exist in a low-energy, stable configuration in which they are around the oxygen atom more often, which results in the oxygen atom having higher electron density and being more negatively charged in comparison to the more positively charged hydrogens.
Though this dipole moment exists within the molecule, it does not bring the atoms out of their current configuration. Hydrogen and oxygen atoms cannot get any closer because their nuclei (both positively charged) would repel each other and increase the energy of the system. And the electrons cannot collide with the nuclei (at normal, low energy environments) because electrons are bound in specific, quantized orbitals around the nuclei and can only exist in those discrete states (which is why negative electrons can orbit positive nuclei without spiraling into them). The attraction between opposite charges definitely exists, but no, the particles will not collide because of it.