Sandra,
If you are using the version that Brad has posted, the "B" in 123B Main St would need to be part of the house number field. [Correct me if I am wrong about that, Brad]
For the version I posted, here is a bit of context...
Our City is part of a regional alliance that happens to have a long-standing address layer schema standard for use in certain regional applications. The schema has its shortcomings and many cities maintain their address layers with fields that meet their individual needs but can still be exported to that regional schema for data sharing.
We have some legacy addressing that includes fractional halves (i.e. 123 1/2 Main St) as well as your example of 123B Main St. While the "B" is really a unit indicator, these legacy addresses have used the 123B format for so long that we simply treat them as fractionals. But we are a reasonably small city and the number of these is small. All newer addressing adheres to the 123 Main St Unit B format.
The regional schema mentioned above is a numeric field, so that prevents me from concatenating the fractional half or letter into the house number. Thus, my version of the locator includes a fractional field as well as the unit field.
For those legacy addresses, the "B" goes into the fractional field and not the unit field. In the locator, a space is required between the 123 and the "B". Otherwise the 123B will be searched against the house number only instead of the house number and fractional fields.
All of that said, if you standardize your "A" and "B" units as 123A and 123B, you are likely better off either making them a part of the house number or using the fractional field rather than the unit field. You really need to proceed based on how you expect users to enter the address for matching, and there is no right answer.
Maybe others have some additional ideas. Apologies for the length but I wanted to provide some context regarding why my version is built the way it is. Admittedly, my primary intent was to create something that worked for us. The fact that others might be able to utilize it was simply a bonus.
I do hope that ESRI begins to provide these locators in the "base code".