How is a cathode ray tube different from beta minus radiation? In beta minus the result is one neutron in the nucleus changing to a proton, plus an electron and an anti-neutrino being sent off. 
The antineutrino is indifferent to our health. So I guess what makes a beta source dangerous compared to a cathode ray tube must be a difference in the kinetic energy of the emitted electrons?
 A: Modest energy electrons (such as those found in CRT televisions and monitors) range out (i.e. dump all their energy and stop) very quickly in dense materials like glass, so these tubes are not emitting significant numbers of electrons and those that penetrate are even lower in energy.
In fact they do emits small quantities of soft x-rays do to electron interactions with the material, but again the rate is low and the energy is minimal so there is little penetration.
You probably don't want to sleep on an operating CRT, but watching television subjects you to a infinitesimal dose. (And the allowed dose is regulated throughout the industrialized world.)

We can use the online interface to PSTAR to quantify the range TVs runs at a few tens of thousands of volts, meaning the electrons get, say 30,000 eV = 30 keV = 0.03 MeV, so the penetration is around $10^{-4}\text{ g/cm^2}$, which given that the density of glass is about 2.5ish we get a range of $4\times 10^{-5}\text{ cm}$.
A: Beta emitters, or indeed most radioactive materials, aren't especially dangerous unless they get into your body. For example iodine 131 (a beta emitter) is concentrated in the thyroid and causes destruction of the thyroid and/or a cancer there. Likewise plutonium (an alpha emitter) is most dangerous when particles are inhaled because they lodge in the lung and cause tissue damage and probably cancer. This is how Alexander Litvinenko was murdered!
Anyhow the electrons in a CRT don't make it through the glass, so even if you pressed yourself up against the TV screen you still wouldn't be hit by any electrons. There used to be sporadic rumours that the collisions of electrons with the glass screen could generate X-rays, but as I recall the X-rays generated are barely measurable and certainly not dangerous to health. If you removed the glass from a TV (and somehow managed to maintain the vaccum) the electrons would kill any tissue you exposed to them, just like iodine 131. I suppose you could kill yourself that way, though I imagine the vacuum would kill you first.
