If you know when the stock was purchased, here are some tips:

• Sign in to your brokerage account
  • Although your broker may not include your basis on your 1099-B, it doesn’t necessarily mean they don’t have it.
  • If your current broker was the selling agent, there is usually a wealth of documentation available in your online brokerage account that can help, such as detailed reports on each sale.
• Look at previous broker statements
  • Review your records for the trade confirmation when you bought the shares. (It's always a good idea to save these for tax purposes.)
  • If you purchased the stock at different times, or haven't sold all the shares at once, you may have more than one trade confirmation statement.
• Contact your brokerage firm 
  • Your broker should have a record of the purchase, if you bought the stock from them.
  • If not, they might still be able and willing to look up the historical stock price for you.
• Go online for historical stock prices
  • For example, the historical section at Marketwatch or Nasdaq. It's generally acceptable to take the lowest and highest price from a given day and average them to arrive at a cost.
  • These free services may not include events that affect basis, such as reinvested dividends, spin-offs and stock splits.
• Go directly to the source 
  • Many companies have an investor relations section on their website which contains historical stock data. Or you can call the company's shareholder services department for help.
  • For shares purchased more than 10 years ago, go to a public library or law school library and look for back issues of newspapers, such as USA Today, to find the high and low price on the date of purchase.

To be sure, for older stock, even this can be a challenge.  Still, if you can come up with a date of purchase, or at least a real good estimate, you can use resources, and even the broker, to determine what the purchase price of the stock was to determine basis.  (If you inherited the stock, you need the value of the stock on the date the person you inherited it from passed).  A broker can give you the stock's history:  when it may have split, etc., that can help to determine the true cost basis of the stock.

The effort is worth it, however, because if you don't come up with a basis, the IRS will determine your basis is zero.  That would mean that the sales price of the stock would be your gain, and that your entire "gain" would be taxable.

This website had a few additional suggestions for how to determine the cost basis you might find interesting: How to Find Unknown Cost Basis of Bonds & Stocks | Finance - Zacks