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.
As bitcoin history shows, currency exchanges will emerge very quickly as soon as the currency is considered to have some nonzero value, despite exchange rate fluctuations. The rates on all these exchanges tend to quickly converge, and are used as a basis even for other people who trade directly. Even after failures of large exchanges like MtGox, the trade went on.

Unlike Bitcoin, you have equivalent goods on both planets (gold, drinking water, rocket fuel, ...), so trader can use them to calculate the estimated value of foreign currency. 

When the interplanetary trade gets non-negligible and it gets regulated, these exchanges gradually enter the regulated regime and precautions to stabilize the exchange rate are made. I assume here both governemnts view the interplanetary trade as something good or neutral, not try to stop it.