Hi! There's no problem using beetroot as a stain (gives a red-purple colour - obvious I know!). Make your batch of stain in one go so that it's consistent. You can then apply as many coats as you need to get the right colour, or add colours on top. If you are making the stain yourself from fresh beetroots add a pinch of salt and one drop of liquid soap as this will break down the surface tension of the water and make the solution easier to apply. It you're using the juice from bottled beetroot - which I think are in vinegar - the acetic acid will probably have a slight effect on the colour, as it will change the colour of wood. However, an change will probably be imperceptable as the strength of the acetic acid in vinegar will not be any where near that of pure acetic acid.
Seal with dewaxed shellac and apply finish of choice. If you are using old fashioned varnish this will also seal.
Hope this helps. Edmund