This information is needed only if you make a nonqualified distribution from a Roth IRA which means that you need to supply this information on Form 8606 Part III. If you do not make any distributions from your Roth IRA before you reach age 59½ there will be no need to know your contribution basis since any distributions from your Roth IRA are qualified distributions, automatically tax and penalty free.
If you make distributions before age 59½ but they are less than what you know to be the amount of your contributions, you again have little problem because distributions are not taxable or subject to early-distribution penalty until your distributions exceed what you've contributed. You could potentially just enter the amount you know, but that kind of locks you in to that, minus the amount of any distributions, being your contribution basis to be applied to any future distributions from your Roth IRAs before age 59½.
If you made a previous distribution from your Roth IRAs you would have had to have provided your contribution basis on Form 8606 for that year, so you might have an old tax return that shows the amount of contributions up to that point.
If you know you have made the maximum possible Roth IRA contribution each year since 1998, the year Roth IRAs came into existence, you might be able to reconstruct your contribution history from knowing that. You could also just estimate your contributions and let the IRS challenge the reporting on Form 8606 Part III if they choose to do so, but that would be a last resort since it might mean that you would be filing an incorrect tax return that results in an underreporting of taxable income.
