The FASP protocol is fully reliable by design, meaning that a FASP transfer session will not complete with a "successful" status unless all bytes in the source file(s) are transferred to the destination file system and successfully written to disk, with a byte-for-byte identical copy of the file data as it was read from the source file system.
If the protocol is run with the transport encryption option enabled, FASP also verifies that the data transmitted was not altered in flight, between the sender and receiver on the network path, by applying a cryptographic checksum (message authentication code) to each packet before it is put on the network and verifying this checksum when the packet is received from the network.
This is described in the summary of the FASP security model
In summary it is not necessary to run file level checksums on files transferred by Aspera products for the purpose of verifying the transport's reliability. However if you would like to run checksums for workflow reasons, two convenient ways to do this are a) to use the post-process scripting capability to execute an md5sum on the transferred files when the transfer completes, or b) to use the manifest file feature to read the files that were transferred and execute an md5sum on each.