Moldings: patch, strip, or replace

Perhaps you want to strip a small area first and see how you like the process and results. There are chemical strippers commonly available. Just tape the area you do not want to strip well.

In my recent experience there is very little ROI doing much of anything with moldings (even though my personal preference is towards them). People are going into buildings known for their beautiful wood moldings and painting them and people are going into units with beautiful ornate moldings and ripping them out.

I am not sure I would look at moldings as an ROI type of improvement. It is not like putting in a washer/drier or upgrading the kitchen. Even if you restore the original moldings, I doubt that is going to be a quantifiable cost improvement when trying to resell down the line.

If the moldings are really bothering you, I would replace them. Restoration work is always a tricky business and costs can quickly balloon depending on what the contractor finds.

Been there done that. Twice.
Option 1 (patch): Don't. You won't be happy with the result for what it will cost you. The labor to make it look passably nice would be better spent stripping or removing.
Option 2: (strip the existing moldings): Worth it, if you like the style of the moldings and casings. : It's an apartment, so unless it was very high end, it's unlikely the moldings ever were natural or stained wood, so you'll be repainting (they could have been faux grained to look like wood - popular in the early 1900s). You may need to do some patching, sanding, or other touch up to the underlying wood, but a good prep job will leave you with nice clean sharp lines, and a new coat of high gloss paint (particularly on the casings) should be a pleasure to look at for a long time.
Option 3 (replace): 2nd most expensive option (after strip), and the contractor may favor it as it's a lot less fussy labor, but could still run some big bucks if you're matching original millwork, most of which is not available at your local lumber company. Alternatively, if the moldings aren't that interesting, you could put in something new with more character.

Sandy,

If you want to keep the aesthetics of the original molding you can have the contractor take a sample and have a template made (last time I did this it cost around $1,000 and then a lumber company can make the exact molding you had before

If you're going to go that way I strongly suggest Dimension Lumber on Stagg St in Brooklyn. Last time I used them they charged me under $200 to cut a knife.