The ralink 5370 should be supported under rt2800usb driver. (Ralink RT2070, RT2770, RT2870, RT3070, RT3071, RT3072, RT3370, RT3572, RT5370, RT5372, RT5572 devices (rt2800usb))
rt2800usb.ko.gz is found in wireless-3.16.6-tinycore.tcz
It's a USB device so you'll need usbutils.tcz installed to find more info. then run from the terminal " lsusb -v | grep -E '\<(Bus|iProduct|bDeviceClass|bDeviceProtocol)' 2>/dev/null "
then post appropriate result, for example:
tc@box:~$ lsusb -v | grep -E '\<(Bus|iProduct|bDeviceClass|bDeviceProtocol)' 2>/dev/null
...
Bus 003 Device 007: ID 0846:9021 NetGear, Inc. WNA3100M(v1) Wireless-N 300 [Realtek RTL8192CU]
bDeviceClass 0 (Defined at Interface level)
bDeviceProtocol 0
iProduct 2 NETGEAR WNA3100M
(Bus Powered)
...
Also please show output of last 25 lines of dmesg eg: " dmesg | tail -25 "