It's easier to read and more informative to use includes and an
itemised CONFIG statement

                        include "P16F628.inc"

will allow you to use the standard Microchip register and bit names

and example CONFIG

    __CONFIG _CP_OFF & _HS_OSC & _WDT_OFF & _PWRTE_OFF & _BODEN_OFF &
_LVP_OFF

shows what is actually set on/off in CONFIG. Without going through the
bit assignments in the include file I've no idea what you want CONFIG
to be doing with 3F10 or 3D10. The problem could be there or with
CMCON. The switch reading loop seems OK

bsf RP0 instead of bsf STATUS, 5 and explicitly use RP1

ie if you must use these as banking controls

rp0 is status,5
rp1 is status,6

then to change to bank1

bsf  status,rp0
bcf  status,rp1

                                            ;comments added !!!!!!

                        bsf        status,5                ;bank 1
(hopefully)
                        movlw   b'11111111'       ;PortA as all inputs
                        movwf   TRISA

;;;;;; Note that unused PortA pins are floating ;;;;;;;;;

If you want just one PortA pin as an input then use movlw 00000010
to set PortA as i/p and the others as outputs

=================================================

You ___MUST___  also set CMCON to get PortA pins as digital

                        movlw   0x07
                        movwf   cmcon

=================================================

                        movlw   b'00000000'       ;PortB as all outputs
                        movwf   TRISB

                        movlw   b'00000100'       ;pre-scaler assigned to
Watchdog ????
                        movwf   OPTION_R
                        clrf    PCON                      ;select 37kHz osc
(if CONFIG is set properly ???????)

                        bcf     STATUS, 5            ;back to bank0
(hopefully)

                        clrf    PORTA
                        clrf    PORTB

; program starts now

BEGIN           btfsc   PORTA, 1              ;wait for a.1 to ground
                        goto    BEGIN
                        movlw   b'11111111'      ;set all b outputs high =
light LEDs
                        movwf   PORTB
KEEPON          btfss   PORTA, 1         ;wait for a.1 to return high
                        goto    KEEPON
                        clrf    PORTB                  ;set all b outputs
low = LEDs off
                        goto    BEGIN                 ;loop

                        END

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