Having trouble keeping up with the amount of ships the AI sends against me.

Hi!

They're taxing their pop sky-high, focus on shipbuilding, and outright buying ships. To win quickly you need to attack their economy, not their ships: econ and morale starbases, and planets. Do that successfully and their stream of ships will quickly dwindle. If they have big cash reserve, try to take some of it away by trading before you attack. Don't start a war until you're in position to significanly cripple their econ in a first turn or two.

BR, Iztok

In preparation for an invasion of the Drengin I've posted spys on 3 of their economic capitals(guess they got them frm conquored civs) and their tech capital, traded them techs for all but 68 bc of their treasury and like 8 of their ships (puny ones) and imposed an embargo. Its just a matter of time until their debt makes it impossible for them to wage war.

The key too as Iztoc mentioned is biding you time. Wait until you have good techs to have beter ships, and enough production to churn out ships in a few turns.

if you are in a position where they attack you, military starbases might help you fight against the waves.

Wait, wait... it MATTERS where you place spies???   

Because of how defense currently works in the game, it isn't too difficult to shut down an entire AI fleet with hundreds of ships using just 1-3 medium ships of your own. You just need on-type defense equal to at least ~70%-80% of the enemy's maximum fleet attack value. (Psyonic Beams help too, since they give your ship enough offensive punch while taking very little space, leaving the rest for defense and engines.)

If you're trying to fight off superior numbers of superior ships, you're obviously in a bad spot. Most likely, you've been drastically outpaced, and the AI has consequently gotten ahold of powerful combat techs. Very likely, they also have a more robust economy and manufacturing capacity... does your opponent have more planets?

Regardless, if you find yourself in this situation, you have a few options.

1) Forget building your low tech ships, they're just gettin whacked, right? Focus everything on getting more powerful ships and weapons, and start cranking them out once they're powerful enough to outclass the ships coming at you. Unless your tech disadvantage is absolutely crippling, this should eventually work.

Keep in mind that the main thing saving you here is that even if your fleet is all but wiped out, the AI isn't very aggressive about landing transports on your planets.

2) Use your existing ships VERY carefully. Don't send them into fights that they'll lose unless you really need to take out the target. Instead of trying to fight the warships that you can't outgun, go for transports, starbases, and traders... every soft target available. This works much better if you're faster.

If you're playing Dark Avatar as the Arceans (with their first strike ability) you can still fight toe to toe with the enemy ships as long as your fleets have a) enough firepower to take out many of the ships in the first volley and you manuever them to have first strike in all battles

3) Ignore the fighting altogether and just use your transports to take his worlds. Defenders tend to be old, outclassed ships, so use your existing military to crush them and take the planets for yourself. If his empire is cut in half and he still thinks he's winning because of his bigger military, well... the joke is on him, isn't it?

It certainly does.
Spy on each planet and find out where there are major developed tiles.
Like the various capitals, econ. tech. etc. and dump a spy on these.