The metric the question mentions is 'flattening the curve of cases'. There is some debate on how to measure this, but the specific metric I've decided to use is to fit a polynomial curve to the graph of cases over time, starting from when each state recorded their 100th case. I then take the second derivative of this curve, and evaluate it at the current date (June 29th), giving us the rate of change of the slope. This is our 'curve-flattening score'.
To give an example, let's look at the graph of cases over time for New York and Texas, shown below. Clearly, in raw numerical terms, New York has far more cases than Texas. One way to measure the handling of the pandemic would be to look at this number. On the other hand, if we look at the trajectory of the curve, New York's curve is flattening off, while Texas appears to be getting steeper. This latter metric is what I will be using in my analysis.
Using this metric, we get the following heatmap, with a higher number representing a worse outcome. We can compare this to the next map showing the party of the state governor. Note that Alaska & Hawaii are omitted in the first map, but are included in the full data below.
Looking at the governor of each state, we can see that the three states with the lowest score are are Texas, Florida & Arizona, all with Republican governors, although these are then followed by the Democrat-governed California and Illinois. Of the worst ten states, seven have Republican governors. Looking at the states judged to be handling the pandemic best by this metric, all of the top five, New York, Michigan, New Jersey, Louisiana, and Virginia are governed by Democrats, and eight of the top ten.
Removing outliers (values outside [-150, 150]), the average Democrat-governed state has a 'curve-flattening score' of -7.66, while the average Republican-governed state has a score of 20.05. This, again, suggests that Democratic governors are handling the pandemic better in their states than their Republican colleagues, based on this metric.
Based solely on this limited analysis, then, it seems that Democratic-governed states are flattening the curve of cases better than Republican-governed states.
Full data:
State Value Party
New York -270.716627 democrat
Michigan -141.942838 democrat
New Jersey -82.264133 democrat
Louisiana -77.507513 democrat
Virginia -39.734256 democrat
Maryland -34.473446 republican
Connecticut -29.163433 democrat
Pennsylvania -21.205778 democrat
Wisconsin -19.795953 democrat
South Dakota -2.257446 republican
Vermont -1.847893 republican
Washington -1.718445 democrat
Maine -1.501176 republican
Alaska -0.931880 republican
Rhode Island -0.076976 democrat
West Virginia 0.298186 democrat
New Hampshire 0.380728 republican
Hawaii 0.660018 democrat
North Carolina 0.816185 democrat
Montana 1.903081 democrat
North Dakota 2.529137 republican
Wyoming 3.486826 republican
Kentucky 5.186839 republican
New Mexico 6.325482 democrat
Delaware 11.543138 democrat
Colorado 12.881185 democrat
Alabama 13.993028 republican
Arkansas 14.869030 republican
Missouri 14.881671 republican
Oregon 16.153599 democrat
Indiana 16.677088 republican
Idaho 18.562333 republican
Utah 19.872332 republican
Minnesota 24.596237 democrat
Mississippi 25.444759 republican
Nebraska 26.300690 republican
Nevada 28.483117 democrat
Tennessee 37.187872 republican
Oklahoma 40.680940 republican
Ohio 41.395639 republican
Iowa 44.830352 republican
Kansas 48.995664 democrat
Massachusetts 49.924742 republican
Georgia 71.282481 republican
South Carolina 74.612648 republican
Illinois 99.889327 democrat
California 187.667507 democrat
Arizona 236.048719 republican
Florida 356.647006 republican
Texas 406.084270 republican