In class we had been taught that Rutherford's model was unsuccessful because it failed to show that the orbits are stable because the electrons would lose energy because of electromagnetic radiation. Bohr had suggested that electrons would move in orbits having well defined energy levels, but my question is that what would prevent these electrons from emitting radiation and collapsing into the nucleus.
Not every model is perfect. Rutherford's model suffered from the problem of electromagnetic radiation but answered important questions about the structure of atoms by showing the existence of a positively charged nucleus.
Bohr improved on this by realising that if electrons could only have certain energies then he could explain other things such as the spectra of hydrogen. As far as I am aware he did not attempt to explain why electrons are fixed in these orbits - this was only explained later by the advent of quantum mechanics. However, this does not mean that Bohr's model was useless since it accurately predicts useful things.
We do not have to know everything about why a model works in order for it to be useful. Einstein did not know why the speed of light was constant but he assumed that it was (based on all the available experimental evidence) and derived special relativity, which is an extremely useful theory. Nowadays we have quantum mechanics and general relativity which we know are incompatible but they are both exceptionally useful theories so we do not ignore them.
As you might have studied Bohr's model includes various orbits. An orbit is a Defined as a particular region in space where the electron moves and has constant energy. A particle that has constant energy will not collapse in the nucleus. He even went forward to calculate these specific distances from the nucleus he came to the conclusion equation mvr=nh/2pi where the terms stand for their general meaning. By this equation we get to know about specific distances from the nucleus where the electron can be present without losing any energy. See the thing is Bohr's model was only created for hydrogen, what you are actually seeing is the adaptations made to the Bohr's model that help us calculate values like velocities, energies, radii, etc. The Logic behind the discarding of the Rutherford model was that the fact that classical electromagnetism states that ACCELERATING charges product EM waves and hence an electron would lose energy.(Circular Motion is a motion where constant centripetal acceleration is needed hence the electron actually experiences the constant acceleration). This constant energy loss would collapse the structure proposed in almost 10^(-8) Seconds as you might have studied. But this does not happen.