It took Kentucky a little more than a month to reach 5,000-plus deaths after surpassing 4,000 deaths on Feb. 6.
The state's positivity rate increased for the second day in a row but remained below 7%.
Beshear later said it was too early to determine if the sites will be able to open Friday.
Sunday marked the 19th consecutive day Kentucky has reported 30 or more deaths in which the virus was a contributing factor.
Wednesday marked the 15th straight day Kentucky has reported more than 30 virus-related deaths.
As of Monday, 1,314 Kentuckians were hospitalized with COVID-19, according to the governor.
Beshear on Monday said that 2,130 of the state's total deaths are attributed to the state's long-term care facilities.
The positivity rate hit its lowest point since Jan. 2 on Friday.
As of Wednesday, 1,678 Kentuckians were hospitalized with COVID-19, according to the governor.
As of Saturday, 1,602 Kentuckians were hospitalized with COVID-19, the governor said, while 410 were being treated for the virus in intensive care units.