Here are two examples that use the Button function:

Manipulate[
 ListPlot[pts, PlotRange -> {{0, 10}, {0, 10}}, Frame -> True, AspectRatio -> 1],
 {{pts, {{5, 5}}}, Locator, LocatorAutoCreate -> True},
 Button["Write points to file", Export["points.csv", pts]]
 ]

which looks like the following:

For your recent example slightly modified one might use nearly the same thing:

Manipulate[Plot[Sin[a x + b], {x, 0, 6}],
     Button["Save a and b to a file", Export["ab.csv", {{a, b}}]],
     {a, 1, 4, Appearance -> "Labeled"},
     {b, 0, 10, Appearance -> "Labeled"}
     ]

which looks like

But your example does not record the user selected value. After the user click the picture, he could not store the point (x, y) values for further use.

Let's say I have the code Manipulate[Plot[Sin[a x + b], {x, 0, 6}], {a, 1, 4}, {b, 0, 10}]

The user selected a=b=1 by sliding the knobs. The user need to store a=b=1 in a file. How can I write this code to let user do it?

Thank you for the reply. What I need to do is to plot the user selected value and let him to see it.

Assume the notebook display two sliders x and y from 0 to 10. The user selected x=y=1. Then he clicked the button. Next he selected x=2 and y=3. Then he clicked the button. There is another button named Draw. When he click Draw, he will see two points (1,1) and (2,3) on the notebook.