Here is an updated version which, I believe, works (note also that I updated the DynamicModule version in the previous poste since it was not quite correct and did not work). Note both the use of the Deployed -> True option so that the content of the Manipulate is not selectable and also adding a Dynamic around the Line graphics directive. In essence I am forcing the Manipulate to behave exactly like the DynamicModule in the earlier (edited) posting. My guess as to why your original Manipulate did not work properly is that it is trying to create internally a LocatorPane. However, the Enabled option to LocatorPane only applies to all of the locators at once: there is no fine-graned use of Enabled in the LocatorPane which can restrict it to a selected subset of the locators. A WRI developer can perhaps comment....

Manipulate[
 Graphics[{
   Locator[Dynamic[p1], Enabled -> move],
   Locator[Dynamic[p2], Enabled -> ! move],
   Dynamic@Line[{p1, p2}]},
  PlotRange -> 1.1,
  Background -> White],
 {{p1, {0, 0}}, ControlType -> None},
 {{p2, {1, 1}}, ControlType -> None},
 {{move, False}, {False, True}},
 Deployed -> True]

As I was moving on, I found another step where I am stuck. I am trying to draw a graph using these points as vertices (actually I will need more points, but I need to understand with only two how things work). I put the graphics of the previous example inside a Show[Graphics...,...] (without the Line[{p1,p2}]), and I am trying to create a complete graph (in this case on two vertices), using the points as the coordinates of the vertices. I added the following code as the second argument in Show

CompleteGraph[2,VertexCoordinates->{p1,p2}]

This indeed produced the graph I needed (the two points connected), but unfortunately, moving one of the Locator points around moves the other one, too. I guess it is about not using "Dynamic" in this line of the code, but no matter where I tried to put Dynamic (before CompleteGraph, around {p1,p2}...) I always got an error message that the two graphics could not be combined. My problem is that I don't understand e.g. the syntax Dynamic@Line . I looked at the Dynamic tutorials, but could not find the explanation for this. Can you please suggest something in this special case and for general reading. Unfortunately I do need to use graphs and not only manually connect the points (which I now could do with Line), because I would like to use built in graph algorithms.

Though I am not sure why you are seeing the behavior in Manipulate that you indicate (it appears to be a bug but others should chime in) here is an example of an implementation using DynamicModule [I have updated this code from an earlier version which did not work properly]:

Deploy@DynamicModule[{p1 = {0, 0}, p2 = {1, 1}, move = False},
  Panel@Column[
    {Checkbox[Dynamic[move]],
     Graphics[
      {Locator[Dynamic[p1], Enabled -> Dynamic[move]],
       Locator[Dynamic[p2], Enabled -> Dynamic[! move]],
       Dynamic@Line[{p1, p2}]},
      PlotRange -> 1.1,
      Background -> White]}]]

And, if you want this to be within a Manipulate, you can do something like this:

Manipulate[

 Graphics[{
   Locator[Dynamic[p1], Enabled -> move],
   Locator[Dynamic[p2], Enabled -> ! move],
   Line[{p1, p2}]}, PlotRange -> 1.1, Background -> White],

 {{p1, {0, 0}}, ControlType -> None},
 {{p2, {1, 1}}, ControlType -> None},
 {{move, False}, {False, True}}]