The Aurora Borealis (commonly known as the northern lights) is a mesmerizing and captivating sight to behold, so beautiful that I just want to touch it and feel it. Should I? Could it harm me?
The lowest reaches of an aurora is ~100 km in the air. Your problem won't be the ionized gas, it'll be that the air pressure is close to zero. Also, aurora are very diffuse, with at most a few glowing molecules per square centimeter. I'm not sure at that density that you could tell you were actually in something.
If you're in a spacesuit to survive the lack of oxygen, you might worry about the high-energy electrons that sometimes make it through to those altitudes, but fundamentally your problem is that you won't really be "touching the aurora" because you're in a space-suit and you can't really see that you're in it.
Even the extremely dim light of the aurora is accumulated from massive volumes of air. This means that a small volume of air emits almost no light by itself. If you were in the middle of a filament you might not even know it! (Likely it would be visible as increased local skyglow, but would be invisible against the Earth.)
Secondly, the aurora are essentially photon emissions from nitrogen and oxygen molecules, so you can't really touch it (as much as you can 'touch' a sunbeam). Even the gas that emits the photons is extremely tenuous. The aurora is emitted between 90 and 150 km in altitude (i.e. mostly above the 'official' boundary of space, 100 km), so ungloving your hand inside an aurora would likely be fatal (unless a fellow astronaut immediately reattaches your glove and repressurizes your suit).
From this document it seems that going to a polar orbit only increases the radiation levels by about 10x compared to a typical medium-inclination orbit, so not enough to be immediately dangerous (but you wouldn't want to put your space station in a polar orbit). Most weather satellites have low polar orbits, I haven't heard of a higher incidence of radiation-induced failures compared to other satellites.
As others have pointed out, you cannot really touch it as such. What you can do, potentially, is to fly through it on a sub-orbital flight. It might be beautiful; indeed, aurora is from a large volume, so you would likely see it all around you.
However, you will be bombarded with charged particles, because that's what the aurora borealis is — the solar wind impinging on the upper atmosphere. So, get some good shielding. Regrettably, in that case you won't see anything.
Virgin Galactic/Spaceport Sweden were planning to fly tourists through the aurora (Swedish language page) in sub-orbital spaceflights. I think they reneged on those plans after talking to local space physicsts, but I'm not sure. If you can lay down the 200,000$ for a ticket, they might take you on a sub-orbital flight anyway.
If you check the Wiki article on auroras, you will find:
The altitudes where auroral emissions occur were revealed by Carl Størmer and his colleagues who used cameras to triangulate more than 12,000 auroras. They discovered that most of the light is produced between 90 and 150 km above the ground, while extending at times to more than 1000 km.
So, touching an aurora is sticking your hand into hard vacuum. It's likely to smart some.
Whether you "should" or not is up to you.