Physics

The US FCC defines ionizing radiation as wavelengths smaller than 124 nm (which corresponds more or less to the ionization energy of both oxygen and hydrogen, so it is a sensible definition).

The "most starlight" part is a bit trickier. Stars emit light in a wide spectrum (approximately black body radiation) depending on their temperature: hotter stars emit bluer light and are more luminous, but very rare, while cooler stars are redder and fainter, and much more common. Yellow stars (spectral type G), like the Sun, emit mostly between 400 nm and 750 nm (visible spectrum), while red stars (spectral type M) mostly emit from 700 nm to 1000 nm,

So let's say that you want all the light with wavelengths of 1000 nm or smaller turned into ionizing radiation. That gives us a blue-shift of 1+z = lambda_obs/lambda_em = 124 nm/ 1000 nm = 0.124.

The relation between speed and blue(/red)-shift is given by the relativistic Doppler effect:

1+z = sqrt((1+beta)/(1-beta))

where beta=v/c and c is the speed of light. Solving for beta

beta = ((1+z)^2 -1)/((1+z)^2 +1)

And plugging the numbers, you get beta = -0.970, where the minus sign means that you are moving towards the star. At 97% of the speed of light.

If you only wanted to turn most of the sunlight into ionizing radiation, you would need "just" 94.7% of the speed of light.