Use your senses
Sight, Sound, and Smell : Unless you are buying from a trusted piano dealer with a good reputation you MUST go and look at the piano or send a competent tuner or person to go and look at it and try it. If you decide to go and look for yourself (recommended), ask yourself the following questions....
How does it look? Is the casework scuffed, damaged, loose or missing? Repairs are often expensive and uneconomical for most pianos listed for sale. The cosmetic condition may be unimportant but will likely be indicative of how well its been cared for over the years. Are there key tops missing - this can be a sign of being kept in an environment that is too dry. Missing key tops can be hard to replace or match if lost. Uneven keys are a sign of moth infestation, as are small coloured particles around the felt components - moth larvae droppings - AVOID!
How does it sound? Even if you are not musical play every key in turn, do they work, do the notes stop sounding promptly when you let go or do they ring on for a long time? Are there any clicking noises or odd sounds? Do the notes sound wildly out of tune - in this case politely decline and walk away.
How does it smell? You may think this is an unusual question but you will be having this very large heavy item in your home, a piano from a home that has been smoked in and reeking of cigarette smoke will bring that aroma into your home and be very difficult to get rid of. Damp and musty smells are indicative not only of damp but also of potential rodent infestation; a piano is a perfect home for mice and vermin.
Other considerations
How much does it cost and where is it? A Piano is a large and heavy piece musical instrument and MUST always be moved by experienced specialists with the right level of insurance cover and capability for the job. For this reason moving the Piano can often cost more than its value, especially if stairs are involved. Nearly all private sellers will expect you to arrange removal so you must get a quote from a professional Piano mover, not a couple of blokes with a van.
Engaging a tuner/technician to go and look at the Piano is a smart move however bear in mind many are very busy and will need to make a charge, this might even be the cost of a tuning. They need to cover the cost of their overheads, tools, insurance, transport etc. A better plan might be to give them as many details, photos as possible of the Piano and ask them for their advice before paying them to go and do an inspection - if there are obvious red flags they will likely advise against even bothering to go and look at it. If the Piano is a serious candidate then paying them to go and look at it will be money well spent, even if its a dud!
Many older Pianos were high quality and were better-built than modern equivalents. A reputable Piano dealer will be able to supply high quality second hand instruments reconditioned or restored to excellent condition.