What moisture content of wood is desirable for building an open veranda?
Relative humidity is different than the humidity present in wood, although it does affect the humidity that is present in the wood.

Since the wood will be outside, the moisture content (MC) of the wood can be around the 15% to 20% range, although the lower the better. 20% MC is getting close to green (fresh cut lumber/timbers). If it is lower than 15%, great. I have measured the MC of wood that was under cover, exposed to the outside humidity and it measured 9% MC, this was in an old storage building with siding. Most framing lumber is considered "kiln dried" (KD) and the MC should be somewhere around 6%-10% MC. This may go even higher if the material is stored for a long time outside. Some timbers, in the US, is sold as green, or seasoned dry and will be in the higher percentages, still good to build an outside structure. If the material will be pressure treated (PT) for longevity, will be VERY wet, more than 20% unless in Europe it is KD.

If the finish product is to be painted, and it is not PT, all cuts are primed to seal against moisture intrusion. The material could be primed before it is cut and used, just prime the ends after they are cut and before it is covered by the next piece or where it joins the next piece.

If the material is PT, the wood needs to have time to loose the excess moisture and have the surface get a chance to leach out the preservative so the paint can bond to the wood, otherwise it will bubble back off. The usual time for PT to "cure" is 6 months before painting, but the ends that get covered do not need to be primed like the standard framing does or should.