I've been playing Starbase 24 quite a bit, as I find it a lot of fun. However, after we destroy the first two waves of enemy ships, they add a new incentive, and your xo/tac officer comes on the screen stating a new mission. Then a whole bunch of enemies warp in from nowhere, and no matter what I do, I always die in that wave. Wherever I am at that moment, there are ships all around me (targetable) and they open fire on me. Sometimes, I don't even last a few seconds before my shields and hull are wiped out.

Yet, others survive this. What am I doing wrong here, and how can I compensate so I'm not always playing this part of the scenario AFTER I rez?

Just make sure you are nowhere near the base after the 50 ships are defeated. That's where most of the new ships warp in while a few groups are near the starting area.

also keep your hull resists high and RSP at the ready. swapping between offence power and a high Aux setting will help this along

The moment they spawn, reverse shield polarity, emergency power to shields, tactical team and Aux2SIF and brace for impact. Or any combination of damage resist stacking and shield/hull healing.

Hit the G.T.F.O. button (Evasive Maneuvers, Emergency Power to Engines, Aux to Inertial Dampers and a Deuterium Tank Burn) and move to the periphery where friendlies are. Especially successful in a cruiser.

Fly an Odyssey and give no craps. XP

In the event you cannot do that, if you're an engi, hit TT1, RSF, EPtS1, Aux2SIF3, , HE1, TSS2, BFI, and EM and get the hell out of dodge. If you're a tac, TT1, APO3, EM, EPtS, BFI and get the hell out of dodge. If you're a sci, Photonic feet, TT1, EPtS1, TSS2, HE1, BFI, and get the hell out of dodge. If you're an engi in an Oddy, put all power to shields, hit TT1, EPtS1, RSF, BFI, Aux2SIF3, TSS2, HE1, and then take it like a man. Then hit BFAW, TS, and give it back.

I keep an eye on the kill counter and start moving away from the starbase towards the spawn point when it's getting close.

If I get caught anyways then I cut and run, full power to shields and engines, evasive maneuvers, and whatever resistance powers I can muster. I also make sure not to shoot anyone on the way out.

It's pretty much impossible to survive a hairy fur ball like that if everyone else on your team has bailed or is exploded.

Meh. I only get overwhelmed after the third or fourth successive combo of csv and torp spread by managing to get 20+ ships attacking me and tac team + engy shield heals can't keep up. I fail to see what the problem is.

The best part is that I can end up dying several times in that instance and still get first place, simply from raw damage dealt.

What are you flying?
Most larger cruisers can soak the damage
if you are an escort either be moving away before the count hits (and ideally Above the action or below it)
If you are a sci spam the hell out of them with lasting effects plenty to choose from

of course a nice blast on the repulsors is always a good option
(Just don't shout "I AM IRON MAN" when doing it )

Engineer in a Fleet Patrol Escort, with two blue mk xi neutronium armor consoles.