How can I hide text cursor in drop down style combo box?

As a best solution, you need to subclass edit control of (drop down) combo box.

To do this:
First, create a MFC class named CCaretlessCBEdit. Select CEdit as its base class.

class CCaretlessCBEdit : public CEdit
{
    ...
};

Then, put lines below into its class definition in the header file to handle the events WM_SETFOCUS and EM_SETSEL.

afx_msg void OnSetFocus(CWnd* pOldWnd);<br />
afx_msg LRESULT OnSetSel(WPARAM wParam, LPARAM lParam);

Then, add following lines to implementation file with message map entries.

BEGIN_MESSAGE_MAP(CCaretlessCBEdit, CEdit)
    ON_WM_SETFOCUS()
    ON_MESSAGE(EM_SETSEL, OnSetSel)
END_MESSAGE_MAP()

// CCaretlessCBEdit message handlers

LRESULT CCaretlessCBEdit::OnSetSel(WPARAM wParam, LPARAM lParam)
{
    LRESULT lRet = DefWindowProc(EM_SETSEL, wParam, lParam);
    HideCaret();
    return lRet;
}

void CCaretlessCBEdit::OnSetFocus(CWnd* pOldWnd)
{
    CEdit::OnSetFocus(pOldWnd);
    HideCaret();
}

Then add following lines inside OnInitDialog() function in your dialog that hosts combo box. Beware of IDC_YOURCOMBO1, replace it with your own.

BOOL CYourDlg::OnInitDialog()
{
    ...

    CDialog::OnInitDialog();

    CWnd* pWnd = GetDlgItem(IDC_YOURCOMBO1)->GetWindow(GW_CHILD);
    if(pWnd->GetSafeHwnd())
        m_ctrlCBEdit.SubclassWindow(pWnd->GetSafeHwnd());

    ...
}

Lastly, put following line somewhere within your dialog class definition.

CCaretlessCBEdit m_ctrlCBEdit;

Dont' forget to include new control's header file in your dialog's header file.

#include "CaretlessCBEdit.h"

PS. As a shorter solution, while handling the set focus notification of combo within dialog class without subclassing the edit control. After the control gets EM_SETSEL message when getting focus for the first time, caret would be still there.