tl;dr: By the time Allomancers were populous enough to be a threat to other people's coin pouches, the economy was too well established for the Lord Ruler to disrupt it. It was a basic risk/reward decision that fell on the side of status-quo.
I could not find an official answer on this from Brandon, though I may try to ask him if I can. However, I think I can speculate based on what we know about Allomancy and how it came about. This is going to involve stuff you don't find out until very late in the trilogy, so spoiler warning if you're still reading.
Based on what Brandon has said about Alendi (the earliest known allomancer), Allomancy as it's currently known didn't exist before the Lord Ruler created Mistborns. We know that allomancers existed, because Alendi was one, but it's not clear where his power came from, or if it worked the same way. It also seems like Allomancy wasn't very well known back then:
Piercings, and Hemalurgy, were part of the world before the coming of Allomancy in its modern form. Then, they were seen as a means of communicating with deity--which, indeed, they were. Ruin manipulated this to make sure any Hero of Ages who came would be under his influence. src - Quote from: VegasDev on October 16, 2008, 01:12:53 AM
This quote implies that people were aware of Hemalurgy -- the form of magic that involves driving metal spikes into people -- but not aware that it gave them powers. We know that Feruchemy was also known, at least among the Terrismen, but they were a very isolated group. It doesn't sound like Allomancy was generally known about at the time.
That means that society would have developed without any threat of Coinshots taking your gold coins and using them for nefarious purposes. The economy would have developed currency for much the same reason it developed on Earth -- gold was rare enough to be valuable, but not so rare as to make it impractical. It was also easy to work with, and didn't rust or corrode or otherwise change it's mass naturally very much.
When the Lord Ruler made his friends Mistborn, he never intended to allow lots of people to gain those powers. There was a limited supply of Lerasium found at the well, and he burned away the mists that like gave early Mistings gained their powers prior to Lerasium's discovery.
Since everyone he made a Mistborn he also made into the noble class, at first the only people with Allomantic powers were the same people who had all the gold. Thus, it probably didn't seem like a big deal, and indeed their ability to use their vast coin reserves for attack/defense was probably a bonus.
As the nobles interbred with the skaa over the years and spread the Allomantic potential to their offspring, more and more lower- and middle-class children were born with Pushing/Pulling ability. By the time that number got high enough to be a problem, though, the "new" economy under the Lord Ruler was well in place. It would probably have been way too disruptive for him to try and replace gold with some other currency.
Metals are Cheap
While I couldn't find anything specifically about currency, Brandon has been asked about a post-Final Empire world where Mistings were more prominent, and what that would mean for society:
Q: 5) A modern world update for a future mistborn trilogy prolly wouldn't involve as much metal, unless mistings were rare, which apparently will not be the case. I'm thinking more like plastics, ceramics, fiberglass, and silicon. I mean for cars and guns and all that.
A: Yes and no. For the rich, this would be an option. But much like using metal weapons in the Mistborn world, it isn't always an option for everyone. You will see both. src - Quote from: Qarlin on October 16, 2008, 02:10:23 AM
The implication here is that the pervasive use of metal all over the place, even when people know better, is simply a matter of expedience. It's easier to make swords -- and money -- out of metal than anything else. If you are very rich, your wealth is in atium anyway, which can't be pushed and pulled; if you're poor, you have to deal with what you have.
So, in the end, the risk of Allomancers using other people's coins was probably just deemed an acceptable risk, compared to trying to impose any of the alternatives.