As is frequently the case with such relatively open questions, the answer is "it depends". What I am going to propose further down is very much my personal view, and others will most definitely have different views.
Probably the safest (and easiest) way to do this is by making the webmail interface of your Exchange 2003 server available to the public. Since your server currently probably does not have a public IP, the easiest way to achieve this is by port forwarding from your gateway or Internet router. I would seriously recommend to only do this for port 443, so you have at least some security in place. You will need a certificate for this. This will only really work, if your internal Exchange server operates on a proper domain (i.e. nothing along the lines of mydomain.local) and your DNS needs to have an entry that points e.g. webmail.yourdomain.com at the public IP of the gateway.
This will also allow users of MS Outlook to connect to the mail server, since Outlook can be configured to use HTTPS connections when the native connection is not possible. This also works for users of Entourage (Outlook for Mac OS X).
I would not recommend exposing your Exchange Server 2003 on the public Internet directly. This product is now 2 generations behind the current release, and most likely MS will stop security updates very soon, if they haven't stopped already. And there are many known vulnerabilities. If you really want to have your MS Exchange server directly on the Internet, I would recommend upgrading and then making sure you have all the security patches installed.