Actually, when dealing with the Alpha particles emitted by most sources, they swill not penetrate beyond the dead layer of one's skin. It is only the soft internal tissues, such as the lungs, where they interact with live tissue and create a concern.
Yes, ingested alpha sources are problematic. A uranium miner is 10-20 times more likely to develop lung cancer than a heavy smoker. With a 2+ charge and their mass, it make them far more ionising than beta and gamma; so DNA comes off second best. Of course, it's no always about ionisation but a combination of ionisation and biochemical effects. Some elements tend to concentrate in the body. Iodine for example absorbs in the thyroid gland, so beta emitting iodine is problematic following nuclear waste release. I am sure you know all this though, but it's good to share ideas and knowledge.
Typically they have a range of 6 - 12 cm in air too. It would be most irresponsible to leave an alpha source on the side and clearly they need to be stored and locked away. However, a person can 'shield' themselves from an alpha source by simply keeping a good layer of air between themselves and the source. It's the same with beta particles, just a bit more air need in that case.