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As most are unfortunately aware the Boeing 737 Max 8 is receiving very bad press after two fatal plane crashes in five months.

China suspends commercial operations of Boeing 737-8 planes

At the time of writing this, not all the 737 Max’s have been grounded. Indeed until only a couple of hours ago Ethiopian Airlines insisted they would continue to fly the aircraft. They have since revised their position.

What recourse will a passenger who is scheduled to fly commercial on that model of plane have if they refuse to board because of safety concerns? Do airlines handle such issues under airline specific ticket contract or there exists some general global aviation rules cover such well founded fears of catastrophe?

well found·ed adjective (especially of a suspicion or belief) based on
good evidence or reasons.

TL;DR

Methinks a well run airline like Ethiopian Airlines and major country like China both deciding to ground the aircraft based on two catastrophic events in a short period of time fits the textbook definition of well founded. Catastrophic plane crashes are not a dime a dozen.

What recourse will a passenger who is scheduled to fly commercial on that model of plane have if they refuse to board because of safety concerns?

None. The airline and the relevant regulators are the competent authorities to determine what types of planes are safe to fly, not the passengers.

Of course, in many cases, airlines will do things they're not obliged to do, because it's good PR. However, I would expect that the collateral "Airline admits passenger who refused to fly on one of their planes might have a valid point" would be worse than the PR of "Airline won't help passenger who refused to fly on the plane that was provided."

If a regulator has ordered that aircraft grounded and an airline continues to fly it, contact the regulator. Beyond that, you would have a hard time arguing that the fears were "well founded". They may not be baseless, but still not "well founded".

At that point, it would be a passenger voluntarily refusing to board. No recourse.

Simple, No recourse. You may always politely ask for rescheduling to different equipment. "It doesn't cost anything to be nice" -- Bear Bryant

NPR.org reports that:

The Federal Aviation Administration says it plans to require a series
of design changes to the Boeing 737 MAX fleet after a pair of fatal
plane crashes including one yesterday in Ethiopia that killed all 157
people on board.

Boeing confirms it's in the process of enhancing the flight control software for the plane, including the MCAS software.

CBC.ca reported that:

The Air Canada Pilots Association, the union for Air Canada and Air
Canada Rouge pilots, sent out a statement urging Transport Canada "to
take proactive action to ensure the safety of the Canadian travelling
public."

Not exactly clear what they meant by that but the Transport Minister (of Canada) would consider it "premature" to ground the 41 planes of that model owned by Canadian airline companies according to CBC.

I'm afraid that doesn't provide further options for the consumer. So I guess all that's left is for people who think the precautionary principle requires them to not fly in such planes would be to cancel their bookings, try to get a refund from the seller and purchase new tickets trying to ascertain what plane they'll be flying on. If as some people have stated it's possible to guarantee that you'll fly in the plane described when booking, then you may choose a direct flight with an airline which doesn't operate the said plane...

Since it is easy to determine what aircraft model is on a particular flight, when you buy the ticket, you are effectively approving that model. So you have no grounds for refusing it and even les grounds for making any demands.