You would need to create some guarantee that someone would be willing to exchange this token for exactly a dollar. This means that the value of the token would in no way be market driven, except that it is pegged to an existing currency and its value fluctuations. I don't believe there is a way to do this in a decentralized manner, but since that's not what you asked, I'll assume that's okay.
You would need to create a company/organization that issues the token, and guarantees that they will buy it back for the price you peg it to. For example, you would always exchange one token for one dollar. This would have to be a legally binding agreement, so that if your organization ever refused to exchange at that rate, holders of the token would have legal recourse against you. You would also need to be able to prove your solvency, and thus hold enough assets in dollars that holders of the token can be assured that you have the dollars to exchange if there was ever a run. If a question of your solvency ever arose, your token would drop in value, so you'd have to allow regular audits of your organization's asset holdings.
Likely, much of this probably negates many of the advantages you are attempting to seek by creating a digital currency in the first place. The trust in your system would be based on the legal system and the government that enforces that system...just like government issued fiat currencies. Your organization would more or less be a bank, but a bank that issues its own token, and therefore is locked out of the existing financial infrastructure. You would be subject to regulation, and therefore won't be able to offer anything more than a bank does.
In the modern age, nearly all money is digital. You've basically just asked how to start a bank.