So What's The Danger Of Owning An Irradiated Silver Dime?

Depends what type of radiation the atricle has been irradiated with:
alpha
beta
gamma
or what part of the electromagetic spectrum.

None of the above.

They are bombarded by neutrons. The Silver 109 atoms capture a neutron and become Silver 110. Silver 110 is unstable and decays by one of the neutrons becoming a proton and emitting an electron (beta partcle) and an antineutrino (Which can pass through light years of lead without reacting with it so I think we can ignore it). This converts the atom into a stable Cadmium 110 atom.

The beta particles can be stopped by the aluminum case of the holder but can penetrate the plasctic cover. But these coins are NOT very radioactive even right after the bombardment and the half life of Cd 110 is only 22 seconds so the radioactivity drops off rapidly. None of these have been made since 1967 so even the most recently made ones are now 1/70,000,000 as strong as the day they were made. Older ones would be even weaker. Frankly I think it would be every unlikely you would even be able to detect anything today.

If the half-life from the bombardment is 22 seconds, the radioactivity would drop by a factor of 1000 every 4 minutes or so. I doubt you could detect any beta from the bombardment even 1 hour afterward.

Today, these things would act more as protective shields against background radiation (by absorbing betas and even some gammas) than as sources of radiation.

(Disclaimer: I am neither a health physicist, nor an MD, so take anything I say with a grain of salt.)