From Maxwell's Equations of Electromagnetism, we know that accelerated charges emit electromagnetic radiation. It can be shown (see here) that the total power radiated by such a charge accelerating with some acceleration $a$ is given by the Larmor formula:
$$P = \frac{1}{4\pi\epsilon_0}\frac{2 e^2}{3 c^3}a^2.$$
Classical electrons are considered to be charged particles executing some form of circular motion, and by definition are thus accelerating. It can be shown using the above formula and some elementary physics (see my answer to this question: Why Rutherford model of atom is unsatisfactory: quantitative estimates) that the time taken by the electron to radiate all its energy would be of the order of $\sim 10^{-11}$s. Keep in mind that it's not just that the fact that the electron would radiate that's strictly a problem: if (by some lucky chance) we had found that the constants of Nature meant that it would take $10^{40}$ years for the electron to lose all its energy, we wouldn't be too worried. It was the fact that it took such a short time, meaning that no atom could ever be stable, that was worrying.
Thus it seemed like the two ideas: the revolving electron and Larmor's formula could not both be true simultaneously. Larmor's formula followed directly from Maxwell's Equations (Purcell has a beautiful derivation of it at the end of his book, Schroeder has a "simplified" version here), so rejecting it would have meant rejecting most of Electromagnetism, so it was much more likely that the Rutherford model was not true.
As to why scientists did not feel that the same thing would apply to planets, I'm not completely qualified to answer, but it seems to me that accelerated masses have no such restriction in Newtonian gravity. In this theory gravity was an "action at a distance" force: if a mass changed its position, the entire gravitational field throughout the universe changed instantaneously, and the resultant gravitational forces were instantly changed accordingly. The changes do not move as waves, as in the case of Electromagnetism.
I'm not an expert, but it seems to me that when we move to General Relativity to describe gravity, such "accelerated masses" do indeed produce gravitational radiation in the form of Gravitational Waves. Note however that unlike the electromagnetic case, acceleration is a necessary but not sufficient condition for such gravitational radiation to be emitted. However I do not know if this was known at the time, and it's quite likely that the amount of radiation would be much smaller than the electromagnetic counterpart!