There exist two free ASIO plugins for Winamp, so far as I know. You can download both of them and see what works best for you.
The first one: You will download a zip archive containing two dll files in a “Bin” folder. Just take one of the two dll files (start with the SSE2) and place it to the Winamp plugins folder. Then start Winamp and select “ASIO output (dll version) SSE2″ as the output option. Press “Configure” and in “Devices” select your ASIO driver – such as ASIO4ALL.
(Note that you can not have ASIO streaming only with a plugin. ASIO Winamp plugins are useful because they let Winamp communicate with the main ASIO driver.)
The second plugin is called “ASIO output Plugin”. This worked with some flaws for me (using the V-DAC of Musical Fidelity), but other people have no problem, as they say. You can try it for yourself. – At least one of these plugins should do the job. Please also note that any file in lower than 44.1 Khz sampling will not even play in ASIO.
Some more advices: 1) do not use resampling, 2) do not force to 16 bit, unless it is necessary, 3) have a large buffer size (unless you are recording and need very low latency).
Especially if you are on Windows 7, check also the Winamp Kernel Streaming plugin, before you use Asio.