Stability of Bohr Orbits
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.