No, this is not true the top tax bracket went from 70% to 50% for tax year 1981, 50% to 38.5% for tax year 1986 and 38.5% to 28% in tax year 1987.
While it is true that these things did happen during his presidency he was not the one responsible for them. All tax bills (which all of this was) start in the house of representatives, are approved by both chambers before being signed by the president. So while he might have signed the he required the help of congress for it to happen. When it comes to tax laws there is no one single person to blame.
All Bills for raising Revenue shall originate in the House of Representatives; but the Senate may propose or concur with Amendments as on other Bills.
Tax Year/Top Bracket
Social security did however start getting taxed in 1984.
Since 1984, Social Security beneficiaries with total income exceeding certain thresholds have been required to claim part of their Social Security benefits as taxable income. The income thresholds for taxation of benefits have remained unchanged since Congress first established them but, because wages have increased, the proportion of Social Security beneficiaries who must pay federal income tax on their benefits has risen over time. In 1984, less than 10 percent of beneficiaries paid federal income tax on their benefits. A Social Security Administration (SSA) microsimulation model, Modeling Income in the Near Term (MINT), projects that 52 percent of families receiving Social Security benefits will pay income tax on their benefits in 2015. Most of these families will be in the upper half of the total-income distribution.