Solution 1:

You can find the equivalent information in slightly uglier form (a.k.a. hexadecimal) in /proc/net/tcp. There, you can find the inode of the connection, which you can look up under /proc/$pid/fd/.

For example:

$ cat /proc/net/tcp
sl  local_address rem_address   st tx_queue rx_queue tr tm->when retrnsmt   uid  timeout inode
 0: 00000000:0016 00000000:0000 0A 00000000:00000000 00:00000000 00000000     0        0 6115 1 f5adc4c0 300 0 0 2 -1
...

(In normal netstat, but not in busybox netstat, the -e option also gives you that extra information.)

You can find the process which corresponds to the inode with the following command:

# for x in $(find /proc/ | grep /fd/); do ls -la $x 2>/dev/null done | grep 6115
...
lrwx------ 1 root root 64  7 jan 22.50 /proc/2560/fd/3 -> socket:[6115]

You need root access for the second step.

Not as convenient as the -p option, obviously, but works in a bind. Could be scripted, if necessary.

Solution 2:

This may not help, if you don't have the opportunity to rebuild Busybox, but in case it helps anyone...

Busybox does have a configuration option to support the -p switch of Busybox netstat. See option CONFIG_FEATURE_NETSTAT_PRG, selected in busybox menuconfig via Networking Utilities → netstat → Enable PID/Program name output.

Solution 3:

If you have or can get ss on your device it can show you the PID too:

ss -ltp # for TCP
ss -lup # for UDP