The reported damage that earbuds are doing (in comparison to older headphones) have concerned me enough that I experimented and plugged an old pair of Philips "behind the head" headphones ($14) into my iPhone for a 4 mile run.
Not only do they work just fine, but the sound (particularly the bass) is dramatically better.
Thread drift wall-of-text inbound... with footnotes too!
Remember though that it's not that ear buds are necessarily dangerous, but rather that they just have the capability
of output audio at ear-damaging volume levels from the power-output of those digital music players. People just never had that kind of power
literally in the power of their hands before -- let alone with ear phones that came with
the player they bought.
More to the point though: any
headphones/speakers can be dangerous for your hearing if given enough power. Don't expect that other headphones types are inherently safer. They may just be less efficient
in producing sound -- or have different response characteristics (e.g. more bass).
The real problem in my mind is not with the ability of headphones to output loud audio -- that's more a choice
matter -- but that cheaper ear-buds lack the ability to block out environment noise.
Environment noise -- the sound of traffic, the engines of the buses, airlines, the crowds of the streets, etc. -- is already loud and can damage your hearing. Add that noise to the fact that you want to hear your music too
and you'll be increasing the volume of your music to ear-damaging levels to raise your music above the noise-floor.
The bass you mention is important too. Lots of times the bass is what people really
want to hear; however, the bass is often what gets drowned out by the environment noise. Rather than using a graphic equalizer on the music player to increase the bass response -- if that setting even exists at all -- most people will just crank the volume. You then hear the bass, but at the same time the higher-frequency spectrum of the music is also increased in volume.
The higher-frequency spectrum is where most hearing loss occurs. The most tragic part is not so much the inability to hear the treble in music, but rather your ability to decipher sound sources. Your ear and brain uses the differences in phase of the audio that it receives to determine where a source of sound is coming from. Without that ability, everything sounds cluttered. You can't tell, for example, which person is talking to you in a room full of guests. Hearing aids don't help much with this either, as they can't capture that phase information and translate it to something your brain can use for triangulation. My grandmother suffers from this (we joke that it was from listening to too much acid trance music in her youth), and she's essentially socially dead in any larger group gatherings as she can't participate in any conversations (having to yell so that she can hear us doesn't help either).
If you find yourself cranking the volume because you can't hear over the noise, try buying some noise-blocking earphones (I don't like noise-cancelling variants though). They cost more than regular ear buds   -- or even a lot
more  -- but you'll be able to hear your music in even the noisiest of environments at very low volumes, and the music sounds much better too. Not only that, you won't be fatigued by the environment noise, such as the engine noise on airlines, or even the engine of your car.
I own  and while some may balk at the price ($225), I've used them for nearly a decade, often daily. They sound great and keep my ears in good condition. There aren't many investments that get used to that extent -- and protect your health. Those headphones can output 122dB of power -- 22dB more than the pain-threshold -- but I rarely need to go above -35dB in the volume  even on airlines sitting near the wing.
 Etymotic MC5
 Shure SE215-CL
 Etymotic ER-4P
 Each 3dB is double the perceived sound volume. The reference 0dB is ~90% of full volume I believe.