What is a reasonable way to determine basis of stocks when the purchase records burned but the purchase month is remembered? Average closing cost that month?

Start by looking through your paperwork for anything that might trigger your memory of when you purchased or received the stock. This could be the actual stock certificate, an old tax return with dividend or other pertinent details, or some event that would have coincided with the stock's purchase, like a Christmas bonus or other unexpected windfall.

Then, try to narrow it down to a date or a date range. Once you have that, follow these instructions to locate historical prices.

If your best estimate is a date range rather than a specific date, use the historical prices at the start date and end date of that time frame to come up with an average stock price for that time period. Make sure you keep a record of your calculation in case the IRS wants to know how you came up with the cost basis.