I assume that you cannot implement another authorization scheme (based on tokens, cookies, etc.) for your API, and you are stuck with basic authentication and its problems . If you use ASP.NET with all the samples available, it can be very easy to add another authentication scheme that is more tailored to web clients (like Office web add-ins).
Having said that, it’s best for me to use HTML5 storage or cookie storage (if not implemented by the browser) to store your credentials.
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