Can you apply a coat of explain paint directly onto a varnish surface? I would like to paint directly onto the varnish pine stair...

I would like to paint directly onto the varnish pine stair skin.
probably but it depends on the varnish type and the gloss type. Sand down the varnish delicately to give a "key" after you can paint a small area to exam.
Gloss paint is not very solid so I would put a coat of undercoat on 1st to cover the wood colour, then you can apply the comment on.
If the gloss is strong colour you may find away with 2 coats of it, but 1 will definately not cover.
You can yet it won't hold up extraordinarily well.
Best item to do is wash the surfaces down near ammonia (I use the instructions for floor wax removal, since some polish residue may be present) I use nylon pads for wood refinishing to scrub the surfaces beside, wipe off the residue later rinse well. This method combines cleaning & sand in one step.
Once dry use an broad-spectrum acylic primer/sealer/bond coat that doesn't require prior sanding (becuase these will stick to soft surfaces). The best I've used are BIN's Bullseye water-base and Benjamin Moore's Freshstart. Both are low oder, cover great (get these tinted to the finish coat)and are easily brushed out. I alway apply two coats and permit dry overnite before caulking any cracks and gap.
AS for the finish paint use a quality acrylic enamel, since these dry faster and smell better, plus are easier to touch-up latter on since acrylics don't "yellow " beside age like alkyds.
If you paint the treads & risers next give those 2 coats of a clear acrylic-urathane after 3 days. Kelly-Moore & others enjoy this for gym floors, cabinets, fauxed surfaces, etc. This will protect the paint from scuff and stronger cleaners.
Answers:    No, at the very tiniest undercoat first.

It is poor paint practise to paint over varnish. The varnish is so hard that paint have a tendency to chip stale when it is banged. When varnish have been applied properly it will own soaked deep into the wood so even when you strip it the surface is usually still hermetic and a wood primer cannot soak in. Hence it is poor drawing practice. (Ever seen an matured white painted staircases with brown chips where on earth the gloss have knocked off)

People do it. (though I never have) Ideally you purloin all or as much as the varnish sour as possible, but that is intricate work. But at least trade name sure you have rubbed them down ample to have keyed/scuffed the entire surface. This is high-status as your paint is not going to properly adhere to varnish and requests those scratches to receive some kind of grip.

You could try a specialist primer first resembling prime-it -plus or Bullseye 123. These primers try to glue themselves to the non-absorbant surface a bit than relying on soaking into it as traditional primers do.
Since adjectives the above answers are a step in the right direction, I would individual add your finish coat will in actual fact survive foot traffic if the primer coat is "thinned" enough to in truth soak into the wood surface. It will stick well if the varnish is etched any chemically or mechanically. The primer should be grease based (mineral spirits) or pigmented shellac (denatured alcohol) Then follow up near your gloss coat(s)!