TXWY AND RWY INCURSION BY A B737-300 DURING A RAIN SHOWER AT SEA.

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THE SEATTLE TO DENVER FLT PUSHED BACK ON TIME FROM THE GATE. AFTER RECEIVING THE SALUTE AND RELEASE FROM GUIDANCE; THE FO CONTACTED THE GND CTLR FOR TAXI INSTRUCTIONS. THE GND CTLR CLRED US TO TAXI TO RWY 16L. I REQUESTED THE BEFORE TKOF CHKLIST; THE FO REPLIED THAT HE DID NOT HAVE THE FINAL LOAD PLANNING DATA; BUT HE WOULD READ THE CHKLIST UP TO THE FIRST LINE. DURING THIS TIME; MODERATE RAINFALL HAD STARTED; THE ACFT WINDSHIELD WAS COVERED WITH LARGE DROPLETS OF WATER RESTRICTING VISIBILITY. THE FO WAS SPEAKING TO THE LOAD PLANNER ON VHF #2. I ATTEMPTED TO CYCLE THE WINDSHIELD WIPER TWICE DURING THIS TIME. I PROCEEDED WBOUND ON TXWY L EXPECTING TO SEE TXWY A AT MY 2-3 O'CLOCK POS. I DID NOTICE THAT TXWY B WAS CLOSED AND BARRICADED. I CONTINUED WBOUND ON TXWY L AND REALIZED THAT I WAS HEADING FOR THE RWY. AS I SAW THE RWY MARKINGS; I IMMEDIATELY STOPPED THE AIRPLANE WELL SHORT OF RWY 16L. THE SUDDEN STOPPING MADE THE FO LOOK UP. I ASKED THE FO; WHICH TXWY ARE WE TAKING. I THEN REALIZED THAT I HAD CROSSED THE RWY HOLD SHORT LINE. SECONDS LATER; THE GND CTLR ASKED ABOUT OUR INTENTIONS. WE ASKED TO TURN 180 DEGS ON TXWY L TO REJOIN TXWY A. THE GND CTLR APPROVED OUR REQUEST AND WE REJOINED TXWY A. CONTRIBUTING FACTORS: 1) LOW VISIBILITY DUE TO SUDDEN AND MODERATE RAINFALL. 2) CONFUSION ABOUT WHICH TXWY TO TAKE TO THE RWY. (THE CTLR DID NOT SPECIFY TXWY A OR TXWY B.) 3) CLOSURE OF TXWY B AND CONSTRUCTION BARRICADES. 4) HOLD SHORT BARS NOT VISIBLE DUE TO HVY RAIN. 5) NO WIG WAG LIGHTS AT TXWY. 6) CLOSURE OF TXWY B NOT MENTIONED IN NOTAMS. 7) OVERCONFIDENCE TO FAMILIARITY OF ARPT. 8) NO SPECIAL INSTRUCTIONS FROM GND CTLR TO TAXI. 9) FO HEADS DOWN DURING TAXI.