Do not download multiple files at once / use different ftp client
You are downloading from TRMMopen (which is an official download source of the NASA / Goddard PMM)
via anonymous ftp. You are only prompted a username/password dialogue if you have multiple connections to this ftp site, e.g. because you are attempting to download more then one file at once.
Instead, try to download one file after another (possibly with a short delay). You won't get the password prompt then.
EDIT: If you used a web browser to download the data, use a decent ftp client instead (which keeps the connection between downloads) . I tested this with a variety of ftp clients, and I had no problems to download larger datasets!
Note that you could contact the data provider and ask for alternatives in order to obtain larger datasets. They offer something like a data subscription service in their FAQ:
"I would like to receive all new files generated for a data product, over a given time period, at a specified time interval (e.g., weekly). Can I set up a procedure by which specified data will be sent to me automatically?"
Yes, just contact firstname.lastname@example.org (or 301-614-5224, voice; or 301-614-5268, fax) and request a "subscription." Information that would be needed from you are data product(s) desired, begin date, and end date. You will receive an email notification whenever data are ready for FTP pickup.
Hovever there are alternative download sources which are probably more suited for downloading larger datasets:
Alternative download sources
The data is available via STORM (Science Team On-Line Request Module):
I highly recommend that you have a look at it:
PPS Data Access - to search for GPM and TRMM data, order custom
subsets and set up subscriptions.
PPS Public Archive - to access GPM and TRMM standard products via online ftp.
Use it to obtain custom, larger subsets of TRMM data at once.
They also provide a different link to the ftp server which also features the 3B42 data. (I do not know if similar connection restrictions apply there though.)
Of course you can also get the data via http from GES DISC mirador, (but this is probably more suitable for single files), or use TOVAS.
Data access policy
Of course, be sure to conform to the the data access policy which can be found in the
3B42 / 3B43 documentation (p.40). Also, have a look at the (quite open) usage policy of the GES DISC.
The different types of TRMM data products have also been addressed in a GIS.SE question about TRMM data download.