Question: There are two worlds of more or less equivalent technological level, say a difference of only a few decades and no more than 50 years. A first contact event occurs and after the hoopla dies down trading starts. 

Initially, barter is the primary form of trade but this quickly becomes cumbersome.

How would the two economies first establish a currency exchange and exchange rate?

Some commodities are similar and occur in similar quantities, particularly minerals gold, iron etc. Other commodities are unique to one civilization or the other; plants and animals unique to each planet, unique tech developed by one and not the other.

Some restrictions:

* Both planets have one currency or at least a dominant currency (like the USD)
* Neither planet is in a "dominant" economic or technological position.
* Travel between the planets takes a few days
* There is a moderate level of trust in business dealings but it is not clear how laws apply to off worlders on each planet.

Ideas I've toyed with:

* comparable commodity-based exchange, ie. gold standard type exchanges, but this seems archaic for technologically advanced worlds and would anyone really want to transport heavy gold out of the gravity well of a planet
* some type of crypto type exchange, but this seems problematic with distance and not very conducive to trade as the price fluctuates

Answer: 
Why have an exchange rate?

Just deal in the local currency on whichever end of the trip I am at, and carry my wealth in the form of goods that will sell well at my next stop. Freight carriers don't like to run with empty holds, it is a waste of resources to sit still longer than necessary to unload and reload, and an even bigger waste to move while empty. 

I leave A with a hold full of cargo bound for B. When I get there I sell my cargo for B coin. I then turn around and buy new cargo using said B coin for my trip back to A. On the A side I do the same thing but with A dollars. If I am doing this regularly, I keep accounts on both ends of the trip in that planets currency. When I am ready to retire, or otherwise have a need to consolidate my wealth, I deliberately skew the relative value of my cargos so that A>B just covers the costs, while B>A lets me extract my savings and bring my profits home.

Money is just a convenient means of representing the time and energy that goes into producing something. It only has value as long as all parties agree on what a unit represents. People on A will never see value in B coins, and B will never value A dollars, because they have a different reference base. The only ones who will see value in both are the people who move between places, and they will naturally define their own formula for relative worth base on what they can do with each currency in it's own place.
Comparing commodities. Money, after all, is just an abstract representation of goods and services. The only reason any form of fiat currency has power is its ability to be swapped for either a good or a service. A way to get an idea at what the exchange rate would be is to just compare a common and frequently used commodity between the two and use it to set a benchmark.

The trick is finding the commodity. Any metal that's used in equal measure would be useful, but suppose one planet is iron-rich and the other is iron-poor? Or perhaps one culture values gold whereas the other values silver? If both planets have the same amount of metals and use similar ways to process it, that would be the best, but that's not always the case.

Services would actually be another good place to start. Figure out the 'living wage', as in how much money would be the bare minimum for living expenses i.e. food, clothing, shelter at a minimal level. Or plot a graph with the respective salaries for necessary jobs - like doctor, farmer, lawyer, etc. (Cue lawyer joke.) Take all that information together and you should get a pretty good idea of what the exchange rate should look like.