Android Browser with proxy
As pointed out by Al's comment above, it's not a question of which browser does support it, but rather of how to enable it. And as Sachin's comment suggests, available solutions depend on whether your device is rooted (more and better possibilities) or not.
As long as it's for the browser only, there are multiple easy solutions. So does e.g. HTTP Proxy Settings give you access to the hidden HTTP Proxy settings the Android system itself has available. For more than browsers, ASProxy might be a choice - but it's not for free. It even provides a by-pass for e.g. local networks - so you can decide which addresses are routed via proxy, and which can be accessed directly.
- Support HTTP / HTTPS / SOCKS4 / SOCKS5 proxy
- Support basic / NTLM / NTLMv2 authentication methods
- Individual proxy for only one or several apps
- Multiple profiles support
- Bind configuration to WIFI's SSID / Mobile Network (2G / 3G)
- Widgets for quickly switching on/off proxy
- Low battery and memory consumption (written in C and compiled as native binary)
- Bypass custom IP address
- DNS proxy for guys behind the firewall that disallows to resolve external addresses
- PAC file support (only basic support, thanks to Rhino)