As already stated, it is the loss of defensive units (large groups of fighters do this too!) that causes popular support to drop. Unless the planet supports the Alliance, anyway. Both of Addict-Ant's suggestions are good.
1. Put down two shield generators. Numerically, it is twice as hard to bomb through, but in the early game, it is worth more than the numbers suggest- getting together enough firepower to blow through a shield is hard, but doubling that firepower can be impossible.
2. TIE Interceptors. They are cheap, you can research them pretty quickly, and in numbers they can wipe out Alliance fighter wings, especially bombers. Early in the game, the Alliance usually lacks sufficient capital ships with bombardment modifiers to take out a shield generator. If your opponent uses a group of ships with bomber support, move a slug of fighters onto a planet nearby- when they move to hit that planet, they either retreat or lose their fighter support, which should mean they cannot break through the shield. This is my preferred method, since you will always need more Interceptors and can quickly turn them into a fighter screen for your own attack fleets later on.
3. Lancer Frigates. Just like Interceptors, except they can retreat after taking out enemy fighters. Any ship with plenty of lasers can do this, so in a pinch, try using a few Carrack Cruisers. Later in the game, Assault Transports actually work very well- your opponent probably won't realize their power until you hit him with them the first time. They are almost as good against fighters as Lancer Frigates, they are fast in tactical combat and strategically (making them ideal for intercepting enemy fleets), and they are fairly cheap, making them useful for "behind the lines" defensive fleets where you cannot afford larger ships. Obviously, against capital ships, they are worthless.
4. Do the same thing to the Alliance. If it is early in the game, they probably don't have sufficient forces to muster two fleet groups. So send your fleet(s) after their planets and start bombing. You should have heavier capital ships than they do, so with a few fighter squads (especially Interceptors) you can probably handle anything short of planet-based weapon platforms. You may not get the same diplomacy boosts, but they may withdraw their attacking fleet to defend against you anyway.
5. If you can, intercept their fleet with one of your own. That can be very hard, depending on where your fleet strength is. But if you force them to retreat, they will know you have superior fleet presence congregated into one location. You can move the fleet around, keeping them guessing as to exactly where it went. Unless they are aggressive, the mere threat of a superior fleet can sometimes be effective.
6. When playing as the Empire, I almost always prioritize my planets. If there is a planet supporting the Alliance but under my control, I may pull completely off the planet. You will take a diplomacy hit, but it isn't as bad as the hit you take when the Alliance liberates the system. Plus you can scrap every facility first to keep it from going to the enemy. Sometimes, I will scrap production facilities, but keep the mines/refineries and troops in place. Then I continually do espionage missions there to catch incoming enemy forces. When the Alliance fleet comes knocking, you can scrap the mines and pull your troops out. If the timing is right, you get the benefit of those extra resources for awhile and still deny them their publicity boost.
This is often the only effective means of "defending" yourself from early Alliance aggression. You can always scrap everything, and unless the enemy fleet is in orbit you can pull your troops off for the smaller diplomatic hit. Especially true if you managed to get a little diplomacy done, making the planet neutral when you pull out.
7. Try putting a general on the planet. This improves the odds that your opponent's bombardments damage your civilian facilities. That swings the diplomatic boost to your favor. I wouldn't do this as a rule; best only as a last resort, since they will probably capture the general.
Star Wars: Rebellion, A Field Manual
"O be wise, what can I say more?"