Syslog process id is saved in /var/run/syslog.pid
You can simply check if a process with this pid is running like this:
if kill -0 $(</var/run/syslog.pid)
echo syslogd is running
echo syslogd is NOT running
You can also enable the mark facility by adding this line in /etc/syslogd.conf:
Beware that the separator between mark.debug and /var/adm/messages must be a tab, not a space.
Then, modify the /etc/init.d/syslogd script to set the mark interval to for example 5 minutes, by adding the "-m 5" option to the syslogd command:
/usr/sbin/syslogd -m 5 >/dev/msglog 2>&1 &
After that, you'll see a new line every 5 minutes in the /var/log/messages files like this one:
Aug 15 05:53:26 bestouan -- MARK --
That will allow you to find out at what time syslogd stops, if it really does.